1. The LP Furnace has NOTHING to do with the operation of your Basement AC system and fan.
2. The Basement AC and ITS Fan is separate from the LP furnace and ITS fan
3. The LP furnace, really need to know the Brand and Model of it.
4. The LP furnace just responds to a 12 volt signal from the wall tstat. Once that signal is received, the LP furnace and burner and Fan are all controlled by the furnace NOT the tstat. You cannot run the LP furnace fan by itself, like the Basement AC system.
5. Tstat sends signal. Furnace fan comes ON. after 20 seconds the Burner lights and the furnace heats the RV. When the RV reaches set temp, the Furnace burner shuts OFF and then the Furnace Fan will continue running until the Burner chamber cools down and then it shuts OFF. Can take from 5 minutes to 15 minutes depending on how hot the chamber is. IF the wall tstat calls for Heat BEFORE the cool down cycle is complete the furnace will relight the burner and start heating again. Doug
PS, The Basement AC and fan are 120 volt operated. The Furnace and its fan are 12 volt operated.
Also, did you replace the wall tstat with the original brand and model tstat?