The way you can test if one of the thermostats is for the AC is to turn both of the to the highest temperature setting. Now turn the AC on. It should only blow room air at this point. Lower one of the thermostats to a temperature below the room or if not calibrated in degrees just slowly lower it and see if it causes the AC compressor to come on. If the AC compressor start let it run until it shuts off the compressor. If the thermostat is in direct line with the air flow that may be you reason for short cycling. If it is move the stat out of the air flow or redirect the air flow.
Generally the newer AC unit have the stat build in the return air flow in the unit itself and no outside thermostat.
If you get not reaction form either stat then I would look for the stat in the AC.
Another reason for short cycling may be a low limit switch on the evaporator that shuts the compressor off if the coil temperature is below freezing. This is a safety to prevent freeze over. If this is the case the unit is low on charge or the coil is dirty to the point it is reducing air flow.
WBCCI 12156 AIR 3144 WACHUNG TAC NJ6
2004 Excursion 4x4
1991 34 ft. Excella +220,000 miles, new laminated flooring, new upholstery, new 3200 lbs axles