Each 110v circuit should have a GFI incorporated into one receptacle on the circuit. For example, the 110v circuit going from near my driver's seat and aft along the street side of my living room, dining room, and into the loo has at least five receptacles with the loo undercounter receptacle having the GFI. I have about five separate circuits through the coach living areas and basement with one GFI-receptacle on each circuit. If two or more circuits do not have power, then your problem is likely not a GFI. Just my opinion. Your problem sounds like it's at a common point... like the inverter, your 110v panel of breakers, or on the circuit going to the switch that controls the power supply origin, meaning the switch between accepting power from either the generator or shore.
My brother has a 2000 Marquis Tourmaline and two years ago on a cold overnight stop in a RV park he had power problems that sound similar to yours. If memory serves me, then all his power problems were corrected by finding and pushing a reset button on the inverter. He did not find in the inverter manual that there was a reset button. Nevertheless, he was advised to look for a reset button on his inverter (by Gerald, I believe it was). So he unloaded his basement to provide access to look closely at his inverter and found a button that, when pressed, solved all his power problems.