Yikes - this thread has so many misconceptions about what heating does to moisture. Fortunately, some other posters have tried to correct these misconceptions.
If you have a ventless propane heater (like a Mr. Buddy) then yes...it will add moisture to the air as the propane burns. But having an electric heater or your factory installed propane heater going won't add moisture to your trailer. As was said, the trailer's propane heater vents to the outside, so that is where the moisture generated by burning propane goes. The electric heater burns no hydrocarbons and therefore contributes no moisture.
The vast majority of moisture in your trailer comes from your own body, respiration mostly, but also via your pores. And then there is showering, dishwashing, cooking, etc. Heating hot water via your gas water heater doesn't contribute moisture directly because it (like your furnace) is vented to the outside. You only notice the increased moisture when it is colder because at lower temperatures, surfaces like your windows and walls behind closed closet doors are cold so the moisture condenses on them.
Propane combustion equation: C3H8 + 5O2 = 3CO2 + 4H2O (Propane + Oxygen = Carbon Dioxide and Water) Of course, incomplete combustion can also result in CO (Carbon monoxide).
Relative Humidity: The relative humidity is the amount of water vapor the air is holding right now as a percentage of what it would be holding if it were saturated. ... If you increase the temperature, however, the amount of water vapor the air can hold increases, so the relative humidity decreases.