Your statement that the GFI sockets are working "sort of" indicates that it's not a GFI problem, but I could be wrong. At this point, I'd look for any loose connections in your electrical power distribution panel (circuit breaker panel). Disconnect from all outside power, use a screwdriver and tighten all connections on the buss bars, the circuit breakers and while you're there, tighten all the 12 VDC connections as well. Reapply shore power and see if you have power to those sockets, if you do, time for an adult beverage, if you don't, then disconnect shore power and pull the GFI's out of the wall and make sure that all the connections are tight and secure, recheck with shore power installed, if good, then the beverages await, if not, then replace first one of the GFI sockets, recheck, and if necessary replace the other.
It sounds like you've got a loose common wire somewhere in the system, hopefully at the distribution panel, but it may well be in a GFI. Most of the connections within the AC side of the wiring are "slice connectors" meaning they simply slide the wire into a "knife blade connector" to cut into the wire insulation and hold the bare wire to the connector. They are "OK" at best and often loosen during travel. Why the industry uses them if they are prone to loosening???? Profits, time to install and cost of the components
In the "profit driven world" cheap is better...... <sigh>