I have an old Tempstar A/C unit that stays on after the blower turns off. I replaced the thermostat in the winter and it works fine with the heater (tuns on & off correctly). I removed the new thermostat from the system & the A/C unit still stays on. The only way I can get it to turn off is to turn the breaker off. This an OLD system so there are no circuit boards.

It could be a defective (contact welded) contactor in the outside unit. That will cause the outside unit to run all the time.

You might have a short in the thermostat wiring keeping the ac on.
There might be a a/c blower relay in the furnace that is defective causing the inside blower to run all the time.