Engine Hot A/C Off Light

I would replace the thermostat , AC delco or other OEM part .

This is going to be the coolant temperature sensor or the wiring going to the coolant temperature sensor. When the computer loses the signal from the coolant temperature sensor it assumes that the engine is overheating and takes steps to prevent engine damage. One of which is to disable the air conditioning because the air conditioning puts an additional heat load on the cooling system. The fact that the temperature gauge on the dash is bottomed out, with the engine at operating temperature, confirms this. Hope that helps! Jim