I'm not sure why you put the other years of the T-bird in.
The 2002 should have a hydraulic fan. The 2003-2005 have and electric fan.
Problems with the hydraulic fan are common. Usually, it is the solenoid valve that determines fan speed that messes up. When it does, the fan only turns at minimum speed, and that is not enough to always cool the engine.
Here's how to test for this. Turn the climate control off and start the car. Note the fan speed and noise level. Turn the climate control on, with the AC button on. The engine fan should speed up greatly and get louder. If it doesn't, then you know the fan is the problem. Usually, it is just the solenoid valve (aka actuator). However, since it is difficult to be certain, most dealers will want to replace both the hydraulic fan motor and the hydraulic fan pump. This is very expensive.
The other possibility is that you have trapped air in the cooling system due to a leak. It is almost certain that your degas bottle (The thing that has the radiator cap on it, by the windshield on the driver's side) has cracked by now and should be replaced. After replacement, the factory bleeding procedure must be followed.
Your comment that it comes and goes points to the fan as the problem.