Unit will not shut off by itself.

The booster motor? Do you mean the inducer motor that helps push the exhaust? If the inducer motor won't shut off when the thermostat is satisfied and everything else does shut off, it sounds like the contacts on the relay that powers the inducer are at fault

Sorry there is no way to convert it to a gas unit.. The only way is to replace the furnace with a natural gas unit. You will need a gas line ran to it and a Flue. and then room to fit everything in that space.

need make and model numbers on both units.

The relay on the outside unit is stuck

What are you wiring? Possibly we can help.

If you need further help, reach me via phone at https://www.6ya.com/expert/thomas_092728000e6acb79

call a service guy

There are a few reasons why your furnace is restarting after it reaches temp. and shuts off. First, if you have a mercury bulb thermostat it needs to be level and the heat anticipator needs set properly. If you have an electronic thermostat then you need to make sure the DIP switch settings inside are set properly for propane. If you have a digital thermostat there are also settings to be made for your type of heating. These settings will be explained in the thermostat installation instructions. Also, Make sure your furnace filter is clean. If this does not help then there are some other things to check which may need to be done by a technician depending on your mechanical aptitude. Hope this helps!

You had the condensate from the furnace come loose look for a pvc pipe coming off the furnace it drains the high efficiency heater. I would really check out your hot water heater too.

there is a "control unit" inside the bottom cover of the A/C on your cealing ... inside there is the control box that controls itself. Has a silver peice bout 2'' thick and the size of a quarter this is your freeze control, if your A/C gets TOO cold and might damage itself it kicks on and shuts down your unit(yes even your furnace) need to have a tech come out and fix/replace it, RV Tech, most A/C repair companies know NOTHING about RV's

put the just the fan on turn off the ac and the heat and just fan on make sure the blower motor works cause thats really the only thing you utilize from the furnace in the summer and let me know
then if so does your heat work fine