Does the electric fan stay running after the car is turned off,until the temp. comes down?

Most often caused by a bad fan relay. This is a complicated system. Read the post below and download the GM wiring diagrams. Then check the fuses and relays. All 3 relays must be in perfect working order for the system to work properly.
Car overheats Ricks Free Auto Repair Advice

I do not know if you have one or two fans. The cars with electric fan radiator cooling had two fans which I owned. One fan works with the temp of the engine. It come on and off to control engine temp. The other fan only came on when the A/C is running. The fact is the electric fans can come on even after you shut the car down if it needs it to control engine temp. If your car is not running hot the fan is probably doing what it is supposed to do.

On most modern vehicles the engine cooling fans are electric. These fans are thermostatically controlled and will continue to run even after the vehicle is turned off and until the engine cools down. If you feel the fan is running an excessively long time after you turn off the vehicle, I would recommend you have it looked at by the dealer.
Good luck.

Not necessarily. The vehicle, in motion, has enough air flow through the radiator that the fan is usually not needed. The fan will run if the vehicle sits and idles for a while and the coolant temp rises since there is no air flow through the radiator. If you are worried about fan operation, there are two things to check. The fans will ALWAYS come on if the A/C is turned on. That is because the fans pll air through the A/C coil in front of the radiator. Air must flow through that at all times or pressure in the A/C lines will get too high. So test that circuit simply by running the engine and turning on the A/C. If the fans run, that part works. Now turn the A/C off. Let the engine run until the temp gauge gets to around 200-220 degrees F. The fan should come on about this time and run until the temp drops to about 190. This cycle will continue to keep the coolant from getting too hot. If the temp gauge gets into the red area and the fan has not come on, shut down the vehicle immediately to prevent engine overheating, and then troubleshoot that portion of the fan circuit.

is the car running hot? some cooling fans never come on going down the road because the air comjng through the front of the vehicle keeps the car cool enough that it doesn't need to come on. but if it is overheating then you may want to check the cooling fan relay.

With the A/C on the coolant fan should run constantly to cool the A/C condenser in front of the radiator. With the A/C off the coolant fan should come on around 210 degrees and stay on until the coolant temperature drops to between 190 and 200 degrees. Hope this helps.

if the car is warm the fans are designed to continue to run untill temp drops to normal temp and then fans shut off. This is a normal operation of the fan nothing to worry about. Now if they stay running for more than 10 minutes you have a problem.

bad fan relay, or bad temp sensor, or bad thermostat if the temp stays low. or bad fan, or bad PCM.

couple posabilities first run car up to temp then turn off...when fan stays running unplug the coolant temp sensor for the fan< there are 2 other is for comp/gauge> if fan turns off replace sensor if not replace relay for cooling fan

The fan relay may be malfunctioning (the fan relay is what actually switches the fan on and off and gets its signal from the ECM, which in turn gets its signal from the temp sensor) testing/replacing the relay and temp sensor would be the best thing to start with,... unless you notice any additional (non factory) wire attached to the power wires to the fan.  Sometimes folks, in an attmpt to fix  an overheating or fan relay problem, attach a ignition switched positive lead from the fusebox or other switched source to the positive poer lead on the fan,.. making the fan run whenever the ignition is switched on.