Burning propane produces carbon dioxide and water, which is one reason to prefer an outside-venting appliance since neither of those products is to be desired in an enclosed space like a camper. It is a good idea, as already suggested, to make sure your furnace is properly vented.
But that's not the end of the inquiry if you're experiencing lots of condensation inside your camper on cold mornings.
Remember how on a hot muggy day you get all that condensation on the outside of your cold beer glass? That condensation happens because the cold glass cools the humid warm air along the outside of the glass. Cool air can't hold as much water vapor as warm air so some of the vapor condenses and is collected as water droplets on the outside of the beer glass.
The same thing happens on a cold morning in your camper. You've been exhaling lots of water vapor all night. Since the inside of the camper is relatively warmer than the outside, the warm inside air is able to hold a significant amount of water vapor. But along the sides of the camper, and anywhere else there's a cold surface, the adjoining air will be cooled and will lose some water vapor that condenses on the camper sides and cold surfaces as water.
You would like to minimize/eliminate the problem short of opening windows & vents. That's pretty hard to do unless you travel with a dehumidifier machine. Go ahead and open windows and vents a bit. Turn on your exhaust fan to low. You're probably heating water for coffee and tea so you want to exhaust the carbon dioxide and water produced by the stove anyway. Exhaust that humid air that's build all night and replace it with fresh low-condensation air from outside.
Our experience is that we'll always have some condensation on cold mornings to be toweled up, but the total amount is greatly reduced by opening a window and running the fan for a while.
Good luck!