You may well have a ground fault somewhere in the RV. Since most campground outlets don't have a GFCI (even though arguably it would be a good idea for them to do so), you don't see any problems when camping.
Narrowing it down by circuit is a good start. Frequent causes of ground faults are the fridge or water heater elements shorting to ground, wiring being wet where it shouldn't be (such as a leaking outside outlet--though that should trip the RV's GFCI), and general wiring problems or errors (things like chafed wires or misrouted ground connections that touch neutrals).
A ground fault is something you want to get corrected. It may lead (especially in combination with some other problem) to a hot skin condition for the RV, which can be dangerous, even deadly.
It is possible for GFCIs to go bad, too, but it seems a bit unlikely to me that two would do so simultaneously in your house, and only when you plug in your RV. If other devices that are thought to be OK cause them to trip, that would be a very good reason to investigate the GFCIs.