A propane furnace does not put moisture into the trailer. The oven and stove do, but not the refrigerator or furnace. Moisture is one of the products of the combustion of propane with oxygen. This combustion occurs outside of the trailer, otherwise there would be a very high risk of carbon monoxide poisoning, another product of combustion. There is a heat exchanger that takes the heat from the combustion, without letting any of the combustion components, including water, in. The heat from the heat exchanger then warms the trailer.
The moisture inside the trailer that you see comes primarily from your breath, which is FULL of moisture. If you cook inside the trailer, some moisture comes from that, but it is primarily your breath. Oh, and if you take a hot shower, but that one should be obvious.