Based on what you are saying I would guess it would be thermostat related since the furnace and air conditioner are (at least on mine) independent of each other. The only commonality is the single thermostat control. Other thought assuming the replacement thermostat was the correct one, is that there may be an issue where some wiring in the thermostat area is either touching another or when the wiring to the new thermostat was made, a wire was placed on the wrong terminal connection.
Would anyone know if on this vintage trailer whether the thermostat wires go back to the A/C unit and then travel to the furnace?
'04 Classic 30' S.O.,'03 GMC Savana 2500,'14 Honda CTX 700