Miata A/C and Cooling issues with questions

[Senor_Taylor]

I agree with Scott. Unless there is some weird edge case that changes my understanding of the AC system, I think these issues are unrelated.

Perhaps running the heat was bringing enough air through the heater core (combined with that air being cooled from the AC) that it kept it from overheating until you switched it away?

I'd start simple with new coolant and confirm there aren't leaks/the water pump is spinning appropriately. Perhaps your belt isn't tight enough or you goobered something up when you were fiddling with the AC?