How do I trouble shoot my thermostat to rule it out?

If you disconnect the white wire (heat), only the fans on the AC have control. The heater should not come on.

If it still comes on, you have a problem other than the thermostat.

You may have a choice on the thermostat to select whether you are using electric heat or gas heat. If set to the electric or heat pump setting it will turn the fan on. I use that setting to control my basement fan when the gas furnace is on.