Electric/Gas Heat Pumps on Class A Discovery

On my unit, the heat pump and furnace are controlled by the same thermostat. Yours is most likely similar. When the outside temp gets to the point that it no longer can put out heat, it will automatically switch to furnace mode and is supposed to switch back when the outside temp warms up. If the heat pumps continued to blow air after it switched to furnace, I’d have the system checked.

There's certainly a connection between switching to gas and blowing cold air when you're out of propane. There's no direct connection between running out of propane and the heat system switching to gas mode, which is to say running out of propane doesn't cause it to switch.

RV heat pumps typically don't operate at outside temperatures below something like 30 - 40 degrees, and automatically switch to using the furnace for heat at that point (or possibly electric resistance heat strips, if you have them on your units). If you don't have propane, the furnace clearly isn't going to produce heat.

Needless to say, the system operating on gas mode will use propane and so eventually cause it to run out if not refilled in due time.