[JRTJH]

Also, on a trailer that old, a previous owner could (probably did) replace components through the years... Your comment about the thermostat fan switch caused me to suspect that the thermostat is not the OEM or even an RV thermostat.... You said that the fan switch has off/auto. RV thermostat fan switches in that era had 3 position fan switches. HI/LO/AUTO.

I'd suspect that your thermostat may be at least a part of the problem. I'd start there, make sure the thermostat is working properly before you start tearing the trailer apart to get to the furnace, which is probably "buried in the bowels of the trailer, under the refrigerator"....