Hot, neutral, and ground wiring codes are there for a reason. Make sure they correct.
That said, the hot skin condition when ground is missing means something is shorting to ground. Even without a proper ground, skin should not be hot. If every thing is working as expected there will be no voltage on the ground and no current through the ground wire. It is solely there in case of a malfunction.
Find the defect and correct it. It may be getting worse.
Shore power sources typically trip two different ways. The first and most obvious is over current. Circuit breakers and fuses trip. Test by turning all RV breakers "off". Next turn main breaker "on". Turn each branch breaker on one at a time.
The second is Ground Fault Interupters (GFI). GFI trip when the current in the hot wire is not identical to the current in the neutral wire. Hot skin means there will be current in the ground wire when it is connected. That means the neutral wire has less current than the hot wire. GFI trips.
Many malfunctions can cause a hot skin. One of the most frequent is an electric water heater element. Many will burn out and short to ground if turned on without water in the hot water tank.
Another is a short in an absorption refrigerator heat element. They sometimes fail just like hot water elements do.
Do the easy things first to trouble shoot. Unplug all 120 volt appliances that have plugs. Unplug the microwave, the refrigerator, the coffee pot, everything. If shore power stops tripping then plug things in one at a time. Wait a moment and add another.
Next trip all circuit breakers in main and sub panels in the RV. If shore power stops tripping then reset circuit breakers one at a time. Wait a moment and add another.
Only hot wires are switched in 120 volt circuits. Neutral wires remain connected at all times. A given circuit may trip a shore power GFI even if the hot wire circuit breaker is off. The only way to know for sure is to disconnect the neutral wire as well. This of course requires more skill and wiring knowledge.
If turning circuit breakers "off" does not indicate the problem, disconnect neutral wires one at a time. Many electric systems have a neutral buss in the service entrance circuit breaker box. The branch circuits can often be disconnected there.
There are other possibilities, this covers most likely causes.
I wish you good luck and happy trails ahead!