I'm no expert on RV A/C systems. But, I think you're right that the fan should stay on if set to Low or High. The only time the fan should go on/off in concert with the compressor is when it is in the AUTO position. Since the AC cools and the fan runs, I would focus on the thermostat wiring both where it connects to the AC unit and where it connects to the thermostat. Hopefully, someone took pictures of the wires connected to the old T-Stat before disconnecting. If so, maybe try putting the old T-Stat back on and see if it does the same thing.