First thing disconnect what you think is a fully-charged camper battery and let it rest for a couple hours, then check the voltage. If the voltage doesn't measure at least 12.6V, the battery is weak.
Another thing you can do is put a test light in between the hot wire of the trailer plug on the truck and the plug on the camper. Might also have to connect the ground. If the light lights, there is something drawing power in the camper. The light won't light if there is no complete circuits in the camper. You could perform the same test with an ammeter to determine how many Amps are being pulled by the camper when it should be "dead."