Hi Cottonman,
Welcome to the Heartland Owners Forum.
The thermostat operates a relay that's typically located in the living room air conditioner. On models with a large air intake, there's a metal box above the intake. Not sure how it's located on models that have intake vents spread out along the ceiling. That relay sends power to the furnace circuit board to turn on the blower, open the gas valve, and fire the ignition spark.
When the thermostat is satisfied, the relay opens and it drops power to the circuit board shutting off the furnace.
Since you've already had the thermostat replaced, the suspect items are: the relay and the circuit board on the furnace. If the relay is stuck, it would be equivalent to the thermostat never reaching your set point temperature. BUT, if you can hear that relay clicking, then the problem is probably the circuit board. It's also possible the gas valve is sticking open, but I'd consider that less likely.