Bought dryer 9/13/03.Model #REX463KQ1,Serial#MP2937105.Two problems.#1 doesn't shut off when open door.#2 doesn't dry clothes the way previously has. Had to turn back 4 times on low setting and finally end up on timed setting to finish when just week before like always dried in 30 min.Load normally dry on auto sense dry high heat was still wet.Had to dry on timed dry heat 3 times.What do I need to buy for parts? Seems like dryer not old enough to have to be replaced.

A fairly common problem! The official instructions are

Before using your dryer, wipe the dryer drum with a damp cloth to
remove dust from storing and shipping.
1. Load clothes loosely into the dryer and close the door. Do not
pack the dryer. Allow space for clothes to tumble freely.
2. Turn the Cycle Control knob to the recommended cycle for the type of load being dried. Use the Energy Preferred Automatic Setting (,) to dry most heavyweight and medium weight loads. See "Drying, Cycle, and Temperature Tips."
3. Select a temperature setting on models with a Temp/Fabric selector. The cycle descriptions include temperature setting suggestions.
4. Push the START button.

Stopping and Restarting
You can stop your dryer any time during a cycle.
To stop your dryer
Open the dryer door or turn the Cycle Control knob to OFE
NOTE: The Cycle Control knob should point to an Off area when
the dryer is not in use.
To restart your dryer
1. Close the door.
2. Select a new cycle and temperature (if desired).
3. Press START

Extracted from
http://www.manualslib.com/manual/511763/Whirlpool-Lte5243dq.html?page=10#manual

Hey Car Guy,.. there are two factors that determine how fast the clothes will dry:
1- air temperature, Is the gas burner is lighting and heating the air? Just stop the dryer mid-cycle and feel the clothes - if they are hot, then the burner/heating part of the dryer is working ok.

2 - adequate air going through the dryer. This is usually where the problem lies. There is a lint filter at the dryer somewhere, accessed either where the door opens, or on the top of the dryer....there may be an additional lint filter somewhere on the exhaust duct. Check for that, and check your air flow where it exhausts outslde. See that the flappers of the backdraft damper aren't stuck closed etc.

If you have good heat and good air flow, then it's doing all it can.

test the push button for open and faulty .However see this causes too.

Clogged Vent
If your dryer seems to run forever, it could be because of a clogged vent or internal ductwork. Your dryer may have an automatic cycle that turns off the dryer when the clothes are dry. It does this with a special thermostat or moisture-sensing system.

Normally, this is what happens during an automatic cycle:

The thermostat tells the dryer to heat until the interior of the dryer reaches a pre-set temperature--say 135 degrees.

When the dryer reaches the pre-set temperature, the thermostat tells the timer to begin advancing. (If there's a moisture sensor, the timer advances only if the moisture content of the clothing is low enough.)

The timer advances until the interior cools, then the thermostat tells the timer to stop advancing, and tells the dryer to start heating again.

This cycle continues until the clothes are dry. But...if the vent is clogged, the dryer may never reach the proper operating temperature, so it doesn't send the signal to the timer and the dryer continues to run indefinitely, even if the clothes are completely dry. To fix the problem, clean the vent and/or internal ductwork.


Heating Element

Your dryer's heating element may be partially burned out. If it is, your dryer still heats, but at such a low temperature that it takes three or four times as long to dry the clothes. If the element is partially burned out, replace the heating element. You can check for an ohm reading but will need to refer to the service manual for proper reading. This sometimes on the tech sheet located somewhere on the dryer. Usually between 9-13 ohms.


Cycling thermostat

Although this isn't a common problem, one of the thermostats that controls the temperature in your dryer may break and cause the dryer to heat poorly. If so, you need to replace it. The thermostat is usually a small, round, black device mounted to an oblong steel plate. The plate is mounted to the internal ductwork . This thermostat usually has four wires going to it. Check the 2 terminals that are opposite each other and are the closer together of the 2. These 2 terminals should have continuity. If not replace the thermostat.

if it works on auto but not on time you have a bad timer,it's not counting down,it's better to use auto anyway,when the clothes dry the dryer shuts off on it's own,this way your not using extra electricity,if your clothes dry in 30 minutes and it's on 60 minutes it will still run after the clothes are dry on auto it doesn't.if you open the door and look around the opening you'll see a tag with the model number,your missing a couple of letters on the model number.

We had the same problem and surprised how rampant the issue was after referring to the internet reviews / troubleshooting, etc. The problem started after about 6 months and the solution came to me after much "headaching" over the problem. There were no 'codes' coming up just the door popping open during the cycle. I purchased a (2) twenty pack of 1/2" diamerter x 1/16" thick N42 Neo Earth Magnets and placed them along the top edge and bottom edge of the dryer opening. Instant fix in less than a minute. The thickness does not interfere with the door opening and the strength of the magnets is obvious. Hope this helps. Fix was less than $10 bucks and no service man, etc.

I got cut short.. and disconnected...so if this is a repeat, please forgive me:
Dryers 101
1. Heat alone does not dry clothes
2. Air alone does not dry clothes.
3. Only dry, heated air passing over and through clothes will dry them...

I recommend the following:
Set the dryer for "Air Dry".
Go outside and check the dryer exhaust.
Confirm that the exhaust is spewing out lots of air.

If it is, then you have a need for a technician to inspect everything - in person.

If it is NOT, the you need to determine the clog - or reason that air is not moving through your dryer, clothes and exhaust system.

ONLY 2 possibilities exist:
1. Fan motor is frozen and needs replacement....or
2. Duct/exhaust is clogged with lint and will not allow air to pass through.

Please check for both, clean/clear/fix as needed.

Try using the tub clean cycle and use a good dose of bleach or vinegear. Here is how to run the tub clean cycle.

• To start the TUB CLEAN cycle:
1. Remove any clothing or items from the washer and close the door.
2. Open the dispenser drawer and add liquid chlorine bleach to the bleach compartment.
NOTE: Do not add any detergent to the detergent compartments. Excessive suds may be produced and leak
from the washer.
3. Close the dispenser drawer slowly.
4. Power On and then press the Extra Rinse button for 3 seconds. (Prewash option will be selected.)
5. Press the Start/Pause button to start.
6. After the cycle is complete, leave the door open to dry around the washer door opening, door seal and
door glass.

You have a venting problem and air is not leaving. Its either kinked or clogged.

Sounds like you have a stopped up vent on your home, when the vent gets stopped up the dryer can't circulate the air properly in the dryer and causes it to get to hot which causes the high limit thermostat to kill power to the element. If you need more help with this visit the link below:
Appliance Repair Guides

There are three possible parts that can cause this.
1. Thermal fuse
2. Thermostat
3. Flame sensor

https://www.repairclinic.com/Shop-For-Parts/a8b6d600503/Model-11072212100-Kenmore-Dryer-Parts

You can check the first two with a meter to see if they are ok. If they are then you should replace the flame sensor.