By the looks of it, it appears you had 2 thermostats - one for heat, one for cool. And then you attempted to wire both heat and cool units to a single thermostat. Correct?
With no wires connected, your right, the compressor shouldn’t be trying to run - and in fact, nothing should be trying to run.
If my first statement is/was correct then each unit has its own control power supply. Most thermostats allow for that by removal of a jumper when each unit has its own power supply.
And (and I’m not sure about the Hunter stat) WiFi stats will typically require a common coming from the the unit control power supply.
If by chance in your trying to parallel the units to a common stat and the two unit control power supplies were tied together - it’s possible that they didn’t like it and some smoke was let out of something.
But that or any of the above don’t explain why the compressor runs on its own. Somehow, someway the compressor relay sounds like it has welded its contacts together internally.
Post the manufacturer and model of your heater and a/c unit along with the model of Hunter stat and this can be assessed a little deeper. Not enough info to tell anything with what you’ve given.
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