With the issue being common to all branches, I'd be looking really close at the breaker box. Remove the shore power and start checking for continuity to the ground bus where you should not have continuity.
If the camper skin has voltage as you say, then either the camper main or house breaker should be tripping as soon as you plug it up. You will need to figure out why you have lost ground continuity. Verify your house plug is ground by checking for 120 VAC with one meter lead in the "Line" slot (shorter of the two) and the other in the ground hole. Absence of voltage on the meter will mean you don't have your ground reference and would explain why the breaker is not tripping. You can do the same thing in the camper as well.
BTW, you're not parked right next to a high voltage power line are you?