As you know the AC needs 120 volt to run. In other words you need to be plugged in or tun a generator.
The furnace uses the 12 volt battery power to run the blower. If you are plugged in the trailer has a built in converter that takes the 120 volt power from the pedestal and converts it into 12 volt power. Most people figure one battery per night of furnace. Obviously this is very subjective and dependent on things like how hot you set the thermostat and how well your trailer is insulated.
A propane furnace does not contribute any moisture inside. The propane flame is isolated in a burn chamber and routed outside the trailer. Otherwise you would get moisture and carbon monoxide in the trailer which would be a bad thing. The moisture in the trailer is caused by your breathing and to a much lesser extent to your sweating. On the flip side a portable propane furnace is vented into the trailer so you will get moisture inside.