On the furnace control board you will see spade connections for the blower motor (main fan) speeds. There should be 1 for Heat and one for Cool. The other 2 motor wires should be on terminals designated as Park. Park means not connected. When you run the Fan only at the thermostat, it usually uses the Cool motor speed connection.
First, with the thermostat set to Fan ON check the AC voltage at the furnace between the G and C (common) terminals. You should see 24 volts, AC. If you don't then check that when you have the thermostat set to Heat OFF that you aren't really turning off the entire system. That information should be in the installation manual.
If you do have 24 volts AC between W and Common, then you'll need to CAREFULLY remove the motor wire connected to the Cool spade connector on the control board. Using the meter, measure the AC voltage between that Cool space lug and the bare metal furnace frame. You should see 120 volts AC.