Short answer is yes you do.
Assuming you have a good 30-20 adapter, experience has taught me that 90% of the time it is a water heater element that has failed, next is the electric element in the refrigerator.
Quick troubleshooting is to turn the water heater breaker off and unplug the refrigerator, then plug the coach back into the GFI and see if it trips.
If the GFI trips with refrigerator unplugged & WH breaker off time to look at power cord / adapter & overall wiring.
If not, turn on the WH breaker and see if the GFI trips. If not plug the refrigerator in and again see if the GFI trips.