We had the same problem on our 2001 Adventurer. Take down the trailer socket and disassemble it. Water has entered through a bad seal at the rear where the wires come in. The water works it's way to the terminals and completes a connection between the terminals.
It took me a couple months to find the problem. When I mentioned it to a Winnebago rep at GNR a few years ago he had heard of the problem with the camera coming on but didn't yet know why it was happening.
It's best to replace the socket. When you reattach the new wires give them a good dose of dielectric grease. The grease is non conductive and will help insulate the terminals should they ever get wet again. Before putting the wires through the back of the socket I would also slide some shrink wrap over them. When the socket is reassembled and in place slide the shrink wrap over the back of the socket and warm it with a hair dryer.
Replacing the socket and making sure the wires didn't get wet again cured our problem. We put on another 60,000+ miles before we traded it off and never had the problem reoccur.
Hikerdogs
2013 Winnebago Adventurer