120VAC problems

Based on what you describe it is highly likely that the campground pedestal had an open neutral or was miswired I.e. hot and neutral swapped. If one of the hot legs were swapped with neutral that can cause a “hot skin condition” especially with an open ground. It is also possible that they had a floating neutral which can also cause a hot skin condition. What is interesting is the burning smell and tone change would likely be caused by a high voltage between neutral and at least one line in your rig. The fact that you have many things not working now would be expected. You can determine what was destroyed by the pedestal by turning off all the breakers in your rig and plugging in to power. If the GCFI outlet still trips then you have current leaking between hot and ground somewhere. A GCFI outlet measures the current leaving the hot leg and what is returning on the neutral leg. If the two are not equal it will trip. If this is the case then you will need to start chasing down where you have an incorrect NG bond. I would check the transfer switch. In separately derived systems such as generators or inverters require their own NG bond. This is commonly done by the transfer switch. Per the NEC a system can have only one NG bond. This is to prevent ground loops or the ground wire being used as a current carrying line. It’s highly likely that when your rig got fried something is shorted between ground and neutral. If your rig doesn’t trip the GCFI outlet with all the breakers switched off that is great news. Next turn on one breaker at a time. For each circuit using a volt meter or non contact voltage checker determine if voltage is present at outlets. It’s best to start with convenience outlet circuits first. Unplug everything first. All you are doing is checking the wiring not the items that are plugged into them. At a minimum you will likely need to replace anything burned up by the pedestal. I suspect that the solenoid in the transfer switch has failed. That would cause the no power situation if it failed open. That would also explain the problems with the GCFI outlet. Once you get your rig fixed I strongly recommend you purchase an energy management system such as from Progressive Industries or Surge Guard. They prevent this problem from damaging your rig or potentially killing you.