I got an '85 trans am that probably needs to have its ignition control module replaced.
How do I replace it? I believe its somewhere inside the distributor but I'm afraid there's a chance I will mess up the ignition's timing if I touch the distributor.

Reason why I think ICM migh be broken:
Sometimes the car starts normally but mostly it doesn't. When it doesn't, it's like the car only runs with a few cylinders and stalls almost instantly and no matter how much gas you give the car, it won't do any good.
Sometimes it starts to work when you let the car cool down for a few hours or keep the battery unplugged for a while.

Spark plugs + cords, coolant temp sensor, fuel injectors and distributor cap + rotor have been replaced with no results.

 

Is the car carb or FI?? Did you replace the cap and rotor also?

 

FI. The cap and rotor have been replaced, but not by me.

 

Timing shouldn't be effected unless you turn the dist while working on it.
Make sure dist hold-down bolt is tight and you should be OK. Pop off the rotor, unscrew a couple of screws holding ICM, unplug a couple of connectors and you've got it out. Be sure to apply heat sink compound to the bottom of the ICM when going back in. It will die quick without it.