your camper draws power just sitting. I think the CO and Propane detectors draw around 1 amp. in a couple of days this can run your battery down to nothing (its not good to run your battery down that far).
Yes you can turn your camper off, and that should isolate the battery of you have a cutoff switch.
I would not recommend leaving it on a battery charger. Lead Acid batteries I would recommend hooking a 15A or so charger to them every month during winter and charge them up. Most small motorcycle chargers recommend not using them on large batteries (something about 1.5a not enough current to fully charge battery correctly.
when camper is plugged into AC, the camper SHOULD charge the battery if everything is working correct. But in the olden days, those build in chargers were pretty much junk and over charged the batteries. in the modern days they got a lot better (but there still is some china junk out there that might last 2 years at best). we would need to know what kind of charging system your camper has built in.
and yes, if your charge wire stays hot all the time, unplug your camper from your truck. no sense is killing the truck batteries with the camper load.