ECU location and replacement?

I just bought a 1998 Toyota Corolla with 139k miles on it. It is in great shape except for one issue: it runs hot (about 3/4 gauge). I had the cooling system flushed to see if it would help with no improvement. When I drove to work this AM I noticed the following:
1- It heats up/gets hotter while driving
2- It cools down a little bit when stopped at lights
3- It cools down significantly when stopped and I blast the heat

I have been going through the coolant flow over and over then rememberer the electric fans. With the car idling in the parking garage, I waited to see if the fans came on. Both fans come on when the temp gauge goes over 3/4 guage. And then cycle off, suggesting the fans are not coming on at the right temperature.

From my reading/research online, I believe simply unplugging the fan temp switch will cause the fans to run constantly in "fail safe" mode. Is this correct? If so, would you agree I should try making the 40 minute drive home with the fans running constantly to see if it resolve the temperature problem and, if it does, the switch is the culprit?

Sometimes when its cold, its works for like 10minutes before it goes off. Have serviced the injector, nozzles and have changed the fuel pump, plugs and fuses. The battery and engine was changed 6months back.

Yes to all the questions you asked.
Yes, keeping slight pressure on the accelerator keeps the engine running. And am able to restart on park or Neutral but it goes
off afterwards and car stops moving.

A Car repair shop has determined that my 1997 Toyota Corolla has four possible problems. The ignition coil, Ignition egniter, Distributor or the ECM Computer. My mechanic has determined that all the Ignition coil, Ignition Enighter, and Distributor are all okay. If the ECM computer the last option does not work what could be another answer to my problem?

My Toyota Corolla 1997 has a hard time starting. My engine is in good shape. The Ignition Igniter, Ignition Coil, Distributor, plug wires, spark plugs, battery, and alternator are okay. My car drives smoothly but when it gets hot like all cars do, the car would cut off suddenly and then I would have a hard time starting the car. I would have to wait like ten minutes for it to cool down and then it will start up and drive again, until hot and then cut off again. If a ECM replacement does not solve the problem, then where else should I look for the problem? Thanks much.

The car is idling fast - it was diagnosed at a repair shop as a faulty idle module. I do not want to put more money in the car, because I would like to trade up soon. Can the car be driven safely with the idle module problem? What could possibly happen down the road?