OFten, there's a thermal switch in the furnace that tells it to turn the fan on once the temp reaches a certain point, then, (sometimes) a different switch to turn it off as it cools after the burner is turned off. Simpler systems just turn the fan on when the burner is turned on and off when it is off. One that turns on with the heat is more efficient, since first you aren't blowing cold air as it starts up, and second, you get the heat you paid for after it shuts off rather than letting it just sit there.
The auto switch position on the thermostat allows the furnace to do its thing. The ON position, bypasses that and turns it on continuously. If you have to jiggle the switch to make either of those things happen, it sounds like it is the thermostat. If it never works right, it could be either the thermostat or the furnace control.
If the thermostat is old, I'd consider just replacing it with a more modern (and likely more accurate) electronic thermostat with a setback capability. This would eliminate the thermostat as the problem, and provide some energy and therefore cost savings on your heating bills if you choose to use the setback function (you don't have to, you can use it just like a conventional, non-setback thermostat).
Your utility company may offer a rebate for a setback thermostat...I'd check with them, and the cost could end up free or very minimal; regardless, it should pay for itself in the first heating season easily.