Did either of the thermostats have a common wire connected? I have seen power robbing thermostats leak voltage down W. I have never seen it happen on a goodman though.
Is there anything else wired to W at the furnace, like a humidifier?
You could also wire the old White Rogers thermostat down at the furnace and eliminate all the wiring. Wire only Rh and W, and see if the problem persists. If it's not the thermostat or wiring then I would agree, the board is likely to cause intermittent problems.
To clarify, the heat stays on or just the fan?