Duo therm air conditionning/furnace thermostat in RV

defective thermostat, relay module, or communication cable between them

sounds like thermostat issue not communicating with relay modeule

Obviously someone wired thermostat or something wrong. What colour wires are going to what terminals on thermostat?

sounds like a bad thermostat

there is a fuse located under the shroud on the roof AC on a circuit board that controls the thermostat to turn on the power to the Air Conditioner replace it it is marked F-1 or, on some F-2 etc.
the air conditioner works on both DC voltage for the thermostat and AC voltage for the compressor

try replacing capacitors

What make and model of AC, as well as Make, model of thermostat ? Need that info for proper diagram.

Can you give me a model number off furnace, so I can be sure of proper instructions?

Sounds like a stuck relay on the unit itself. Maybe there is a short in the control wire that comes from the thermostat that activates the fan.

The thermostat probably has one hot wire at all times. The thermostat then chooses which wire to send the voltage down based on your selection. One of those wires is probably for the fan only. Make sure that it is in good condition (no bare, pinched or wires)