If the house GFCI is tripping, it's probably because there IS a ground fault downstream of it and it's not "extra sensitive" or faulty.
It's not impossible that it could be the 20 amp GFCI, but I would not immediately suspect it. You could try running an extension cord to another GFCI recept. in your house and see what happens when you try bringing on the breakers in your RV.
Remove the electrode in the HWT and see if you can bring everything on by turning breakers on. If the GFCI still trips, pull the plug on the fridge as mentioned. A failed HWT tank element or heater element in the fridge are common causes of a GFCI tripping. Until you plug the shore power cord into a GFCI receptacle, you wouldn't know. There's other possibilities like an AC unit or inverter if you had one.
Leave the 30 amp main on with all branch breakers off. Turn each 15 (and 20 for AC) on and off until you find the ones tripping the house GFCI. You *should* be able to isolate the branch circuit(s) causing the ground fault and then narrow it down from there.
There could be a ground to neutral connection somewhere, like in the panel, but since you say that the GFCI you used in your old house was okay, it's likely not this. If you had a volt-ohm meter )VOM) you could see if there is continuity between neutral and ground at the shore power plug.
For the record, GFCIs in series won't cause one of them to trip and this can be ruled out. They will "compete" with each other and you won't know which one will trip first when there is a ground fault