It is the ECU/ECM that sends power to the fan control module and tells it to turn on/off, with the purple wire. It is probably written in C rather than F inside the code of the ECU, the USA being the only country left that uses F. In any case, that will not have a bearing on your problem.

If you have an infrared temperature gauge, heat up the engine and check the coolant temp. The engine fan should come on at 95-100C, which you can check at the top of the thermostat or even the large coolant hose.

I forgot, it is getting warmer there - I just had to shovel 8" of snow off the driveway - so your AC should be warm enough to kick in and start blowing cold air. The engine fan should come on within a few minutes.
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Keeping it running is better than buying new