The LED on the dash does not mean that the compressor is running, it just means you have the A/C system turned on and the compressor will run when it needs to (or at least, it should). If the pressures in the A/C system are incorrect, it will not run or won't run for long. I'm not sure about GMs system, but it could very well be related to the engine temperature. If the computer doesn't know the temperature of the engine, it doesn't want to allow the A/C to kick on and give the engine an increased load.

 

Your probably right about the temp sensor. I'll take a look at it this weekend. Maybe just a loose connection at the sensor. Or even a bad sensor would be alright. While I'm at it, are the temp sensors on these cars located on the intake near the water pump end of the motor?