Hi,
The idea that the compressor was engaging with the A/C off is interesting and could be a relay issue. However, that wouldn't cause it to smoke. When you tried turning the A/C off and the compressor still tried to engage, could you have had the system set to the defrost setting? I ask because in the defrost mode, the compressor is actuated to dehumidify the air entering the vehicle. Try turning the heater on and the airflow fully on the floor setting to see if the same thing happens.
Next, when you saw smoke, did it smell like it could be something rubber burning? I ask because the compressor clutch, or compressor may be locking up and causing the belt to melt/smoke and smell like something is burning.
As far as the A/C relay is concerned, no special tools will be needed to remove it. It is located in the under-hood fuse box. I attached a pic below showing its location. Simply pull straight up on the relay and it will come out. If there is a different relay with the same part number, switch them to see if it makes a difference. If there isn't,
Here is a link that you may find helpful:
https://www.2carpros.com/articles/how-to-check-an-electrical-relay-and-wiring-control-circuit
I question if the relay is the issue because if it was sticking on, the compressor would be engaged regardless of if the HVAC system is on or off. You indicated with the fan off, it didn't engage. That's what makes me question it. I think the belt was burning on a pulley that had excessive resistance or was locked. However, anything is possible.
Let me know if this helps or if you have other questions.
Take care,
Joe
See pic below.
Image (Click to make bigger)
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Wednesday, September 14th, 2022 AT 7:55 PM