RPod 171 Select heat on T-stat, but AC starts up

As noted, when I set the thermostat in my 2013 R-Pod 171 to "heat," the heater does not start but the overhead AC does, and it gets cold quickly, the opposite of what I need.  Since the original thermostat was showing its age by partially obliterating its display, I switched it out for a new Dometic thermostat, hoping that the replacement would solve the confused HVAC problem.  Not to be--the problem persists.  The fan setting starts the fan, the cool setting starts the AC, and the heat setting also starts the AC.  As far as I can tell, the only HVAC controller is the thermostat itself but what's happening suggests a problem elsewhere, perhaps the wiring?  Any insight would be much appreciated.

Edited by sjbarry - Yesterday at 2:04am