My 92' Covette cooling fans start up about 10 seconds after I start up the car cold. Then the check engine light comes on. After about 7 minutes, the fans turn off and then the check engine light goes out. The rest of the day the car acts normally, until the engine gets cold again. What is causing the fans to come on so early??

read code and possible overheating problem

The most common cause of this problem is that the drivers seat heater grid is open or burned out. That is based on 25 years as a Ford dealer technician.

Well it could be a problem with the fan relay, or it could be a safety feature of the computer.
if the computer finds a problem with the cooling fan or the engine temp sensor, it will turn on the fan to try to prevent overheating. Even if the engine is cold.
If the check engine light is not staying on, I would check the fan relay for power and ground.

First thing to check when the engine is cold is that if there is enough antifreeze in the radiator and after you run the engine for about 5/10 minuter check both the rad hoses to see if they are both at the same temperature, if the return hose which is the bottom one is cold then replace the thermostat. If both hoses are hot or same temp then make sure the radiator fan comes on?

A cheap cooling sensor might not start the fan. Make sure the one you or your mechanic bought is a quality product.

Do the following:
1. Clean the throttle body from inside.
2. Adjust the spark plugs gap to 1.00 mm.
3. Check fuel pump pressure. (44 Psi - 49 psi)
4. Remove the negative battery terminal for 5 minutes.
5. Connect the battery terminal.
6. Check the engine oil level and coolant level.
NOTE: Before starting the engine shut off all lights and loads for example room light, blower, radio, parking light etc.
7. Start the engine and hold the throttle at 3000 rpm till radiator fan comes on.
8. After the fan comes on release the throttle pedal and let it idle for 10 minutes.

The cooling fan is made to come on when the engine reaches a certain temperature. To give a simple and general test - turn the AC on high - it your AC and fan are both working correctly the fan should come on almost immediately. If not within very few minutes like 5 or 8.
Also at operating temperature the fan will run as well. In the early morning before you start the car open the hood - fill the radiator while the engine is still cold. start the engine and wait about 10 minutes or so. The fan should come on before the engine over heats. This may take up to 20 minutes and the fan is designed to run after the car is shut off IF - the temperature is hot enough.

Hi it sounds like you have a temp sender playing up I think there are 3 on your engine,

test

no fan comes on when engine cold start up .but it will come on when you turn on air conditioning or defrost on .if engine idling a while when temp get 190 degree the fan will kick on.if over heating you got air in system .wait until cools check coolant add more .crank engine let it idle few. minutes when start warming up turn off engine wait a while before removing radiator cap so you wont get scald.add more coolant .crank it again starting warming up turn off engine wait a while .add coolant .keep doing this procedure until your coolant stop dropping off .if you do this more than 6 times you got a leak some where.check crank level for signs.if oil level higher than normal.if everything looking good.when engine cool make sure add coolant to over flow jug cold level mark.