I agree with Gordon, As to the problem with the furnace, if it turns on from your thermostat, and you can turn it off that way, it is pretty safe to say that the thermostat is controlling the furnace. If you have access to the furnace connections, you could test it by lifting the leads from the thermostat while it is in operation to simulate the opening of the thermostat signal to see if it turns off. I am quite sure that it would.
The signal goes from the furnace, to the thermostat, where it is turned on/off, then from the thermostat back to the furnace, where it passes through the motor relay, and also connects to the sail switch, from there through the high limit device and to the circuit board. That is the power supply to the circuit board which opens the gas valve and sends a voltage to the sparking device or igniter. All that the thermostat does is to make or break that voltage.
I would be pretty comfortable in suggesting that you need a new one.
If Ain't a Newell, It Ain't Wurt Oonin!