A/C blowing hot air and temperature gauge issue

Welcome to 2CarPros.

Okay, there is a chance these are related. So, lets start with the coolant temperature sensor. See pic 1 for sensor location and picture 2 for a legend of components. I did circle the sensor in picture 1 to help locate it.

Now to check the sensor, you will need to remove it and place the sensor end into either hot water (at a known temperature) or hot coolant and check for resistance in the sensor based on the coolant's temperature. Here are the directions.

Inspection

1. Turn the ignition switch OFF.
2. Disconnect the ECTS connector.
3. Remove the ECTS.
4. After immersing the thermistor of the sensor into engine coolant, measure resistance between the ECTS terminals 1 and 2.
5. Check that the resistance is within the specification. See pic 3

I noted you are a mechanic, but am adding a couple links in case it's something you need.

https://www.2carpros.com/articles/how-to-use-a-voltmeter

https://www.2carpros.com/articles/how-to-check-wiring

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Check this and let me know the results.

Take care,
Joe