All dash instruments operate except the digital odometer. Is the problem with the cluster and does the odometer have its own separate motor?

Your first step should be to have the vehicle hooked up to a factory or professional scan tool .
Diagnostic Trouble Codes (DTCs)
The DTCs when set, indicate that the instrument cluster has detected a malfunction in a particular circuit or other system. The instrument cluster is programmed with routines for internal tests that the instrument cluster follows only under prescribed conditions (test conditions). When these conditions exist, the instrument cluster tests certain circuits or systems for a malfunction (failure conditions). When the failure conditions are true, a DTC is set current. Some DTCs command the instrument cluster to illuminate the SERVICE VEHICLE SOON indicator.
The DTCs are not stored in the instrument cluster's electrically erasable programmable read only memory (EEPROM). The DTCs are not retained after a battery disconnect.
Instrument Cluster EEPROM
Important: The EEPROM is soldered into the instrument cluster. You can not service the EEPROM separately.
The instrument cluster contains a data storage device that can save critical data even when battery power is removed. This device is called the electrically erasable programmable read only memory (EEPROM). The EEPROM is used by the instrument cluster in order to store the following information:
• The odometer readings
• The options settings
• The manufacturing information
Speedometer and Odometer
An electronic circuit operates the speedometer. The vehicle speed sensor (VSS), located in the transaxle, generates an ac voltage with a frequency proportional to the speed of the vehicle. The voltage goes to the powertrain control module (PCM) and then to the speedometer circuit board in the instrument cluster. The solid state unit drives the pointer of the speedometer.
The same buffered speed signal from the PCM is processed by the speedometer circuit board in order to drive the odometer. The odometers are operated by a stepper motor that responds to pulses from the speedometer circuit.

Odometer
The odometer is an electronic digital display. Digits displayed are illuminated by the odometer backlight bulb. The odometer backlight bulb brightness is controlled by class 2 messages from the BCM to adjust the display for daylight and instrument panel dimming operation. The trip odometer and total odometer will show in this display area. The push button in the lower right side of the speedometer is used to toggle between the trip and total odometer readings. To reset the trip odometer, push and hold the button down for at least two seconds. The total odometer display has six digits with no decimal display. The trip odometer has four digits including a decimal point and indicates tenths of a mile or kilometer.
The body control module (BCM) stores the vehicle mileage. If the BCM fails, the cluster will display the last valid mileage that was sent before the BCM failed. For replacement of the BCM refer to the BCM section of this service manual. Replacement instrument clusters do not require odometer pre-setting before installation. If the instrument cluster is replaced, the mileage stored in the BCM will be sent to the new cluster via serial data, and shown in the odometer display.

Your best bet , take it to a qualified repair shop.

This is one guy's advice: Replace the speedo head, that should take care of the problem."

This site is also interesting because the guy said
General Motors cars had instrument cluster problems from 2003 through 2006. When I researched instrument cluster problems on the Internet what I found raised concerns in me. Many of those vehicles had problems that began with the instrument cluster, but eventually led to the engine refusing to start.
(an electronic instrument cluster, digital instrument panel or digital dash for short, is a set of instrumentation, including the speedometer, that is displayed with a digital readout rather than with the traditional analog gauges. Many refer to it simply as a digital speedometer.)

http://www.instructables.com/id/Erratic-Instrument-Panel-on-GM-Cars/

After I bought my '96 850, I found it had the same problem. In my case the speedometer worked fine, but not the odometer. A little research on the internet revealed that it's a common problem with these models.

There's a motor that drives the odometer separately from the speedometer, and the weak link in the chain is a tiny gear that turns the odometer when the motor runs. That gear is easily broken and even one missing tooth will disable your odometer.

The fix is relatively straight-forward, but requires you to remove the instrument panel and disassemble it in order to get to the motor housing and the broken gear. You won't have any trouble finding replacement parts on eBay that are much better made and will last you the rest of the life of the car. The difficulty level on this is about medium, but be careful when you've got the instrument panel disassembled; I wasn't cautious enough and ended up with a couple of my indicators out of alignment; my tachometer read zero when it was idling, and my fuel gauge only went up to 17 gallons when it was full, so I had to take it all apart again to reposition the gauges to get them to read correctly.

The GEM has nothing to do with it !
Odometer/Trip Odometer
The odometer and trip odometer are displayed in the integrated circuit LCD display. The instrument cluster records the total mileage of the vehicle using information received from the PCM, and electronically displays this information in the LCD. If this information is not received, the LCD displays all dashes. How did you prove that instrument cluster isn't bad ?
Did you have a factory scan tool hooked up , check for DTC'S - diagnostic trouble code's ? View data to the instrument cluster etc...
Instrument Cluster Diagnostic Trouble Code (DTC) Index DTC Description Source Action B1202 Fuel Sender Circuit Open Instrument cluster Go To Pinpoint Test B . B1204 Fuel Sender Circuit Short to Ground Instrument cluster Go To Pinpoint Test B . B1205 EIC Switch-1 Assembly Circuit Failure Instrument cluster INSTALL a new instrument cluster. REFER to Instrument Cluster in this section. TEST the system for normal operation. B1342 Instrument Cluster Is Defective Instrument cluster CLEAR the DTCs. REPEAT the self-test. If DTC B1342 is present, INSTALL a new instrument cluster. REFER to Instrument Cluster in this section. TEST the system for normal operation. B1359 Ignition Run/ACC Circuit Failure Instrument cluster Go To Pinpoint Test A . B1317 Battery Voltage High Instrument cluster Go To Pinpoint Test N . B1318 Battery Voltage Low Instrument cluster Go To Pinpoint Test N . U2199 Invalid Data for Engine Coolant Instrument cluster Go To Pinpoint Test D .
Your best bet ,take it to a qualified repair shop.

• The speedometer/odometer is inoperative

• Powertrain control module (PCM).
• Instrument cluster.

• Go To Pinpoint Test F .

This solution will be of great help to you,even in the case of future or other dashboard problems;
Instrument cluster consists of electronic analog gauges, various warning lights, and optional tachometer. Vehicles equipped with tachometer have an electronic vacuum fluorescent transmission range indicator (PRND3L),odometer and trip odometer display. Vehicles without tachometer are equipped with a cable-operated transmission range indicator.
Instrument cluster is equipped with the following warning lights:
Battery voltage.
Cruise (speed control).
Door ajar.
Fasten seat belt.
Lift gate ajar.
Low fuel level.
Low windshield and washer level.
Traction Control.
The Body Control Module (BCM), which receives input from various sensors, sends display information to instrument cluster and operates gauges. Instrument cluster incorporates a self-test diagnostic feature to identify electronic problems. BCM, Powertrain Control Module (PCM), and Transmission Control Module (TCM) are continuously monitored by the diagnostic system.

SELF-DIAGNOSTICS
With ignition off, press and hold TRIP and RESET buttons while turning ignition switch to ON position. Continue to hold TRIP and RESET buttons until odometer window displays CODE (approximately 5 seconds). If problem exists, system will display diagnostic trouble codes. See DIAGNOSTIC CODE CHART table. If no problem exists, Code 999 (end test) will be displayed.
DIAGNOSTIC CODE CHART
CodeDescription
110.......Cluster Memory Fault
111.......Cluster Calibration Fault
905.......No CCD Bus Message From TCM
921.......Odometer Fault From BCM
940.......No CCD Bus Message From PCM

After diagnostic code display ends, instrument cluster will go into each of the following tests in order:
Dim test.
Calibration test.
Odometer segment test.
Electronic Transmission Range indicator segment test.
Dim Test
CHEC-0 will be displayed in odometer window. Instrument cluster will dim Vacuum Florescent (VF) display. If VF displays do not dim, a problem in instrument cluster exists. See BODY CONTROL COMPUTER TESTS in BODY CONTROL COMPUTER article.
Calibration Test
CHEC-1 will be displayed in odometer window. Each of instrument cluster gauges will move sequentially through each calibration point. See GAUGE CALIBRATIONS table. If gauge needle is not calibrated properly, a problem in instrument cluster exists.

Odometer Segment Test
CHEC-2 will be displayed in odometer window. Each segment of odometer will light sequentially. If any segment does not light, replace instrument cluster.
Electronic Transmission Range Indicator Segment Test
CHEC-3 will be displayed in odometer window. Each segment of electronic transmission range indicator will light sequentially. If any segment does not light, replace instrument cluster.

you would have to pull the IC(instrument cluster) out of the dash and check the plugs, i have only seen it once where the plug had come out, it is possible here, otherwise you will need to replace the cluster.

Section 13-01: Instrument Cluster—Conventional 1995 Thunderbird, Cougar Workshop Manual REMOVAL AND INSTALLATION Instrument Cluster Removal and Installation

1. Disconnect battery ground cable (14301) .

1. Remove screws holding instrument panel reinforcement (017A28) .

1. Remove screws holding instrument panel lower shield (045G02) .

1. Remove fasteners holding steering wheel column as outlined in Section 11-04 . Gently lower steering column.

1. Remove fasteners holding the instrument panel finish panel (044D70) and gently pry off.
   
   
   
   

1. Remove four instrument cluster retaining screws. (Do not remove screws from the instrument cluster main lens (10887) and instrument cluster mask (10890) ).

1. Slide the instrument cluster (10849) to the right of the instrument panel opening and unhook the instrument panel harness from the back of the instrument cluster .

1. Remove the instrument cluster from the instrument panel (04320) .

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Speedometer/Odometer Federal law requires that the odometer in any replacement speedometer (17255) must register the same mileage as that registered in the removed speedometer . Since the speedometer mileage cannot be reset on this unit, affix odometer sticker to driver's door pillar and record original vehicle mileage. Removal

1. Disconnect battery ground cable (14301) .

1. Remove the instrument cluster (10849) as outlined.

1. Remove seven screws retaining the instrument cluster mask (10890) and instrument cluster main lens (10887) .

1. Remove speedometer assembly.

Installation

1. Position the speedometer assembly to the instrument cluster back plate (10848) and install.

1. Position the instrument cluster mask and instrument cluster main lens to the instrument cluster back plate and install the seven retaining screws.

1. Install the instrument cluster as outlined.

1. Connect battery ground cable and check the operation of the speedometer .

please rate if this helps

It may be possible to get instrument cluster to do a self-test? If you want to try this to rule out instrument cluster.
Press and hold odometer reset button, turn key from off to run, run is where the dash lites come on. Any stored 3-digit codes will be displayed in the odometer vf display. The vf display will perform a sequential segment test. All indicators, data controlled, will illuminate. All gauges will cycle in 12.5% or 15% increments. The cluster will automatically exit self-test. The self test does not verify serial data stream. The cluster internally commands all non-hardwired gauges and indicators to operate. The fuel level sending unit sends info to pcm then pcm sends data to instrument cluster.
I don't know if that self-test will work on your vehicle? It won't hurt to try. If all the gages cycle except fuel gage, then I suspect the instrument cluster. You can call around different shops, get an estimate for the repair.