One of the first things any RVer should do, prior to plugging in, is to check the wiring on the pedestal. Many older parks and even some new ones that have had "repairs" done to the pedestals by uncertified electricians will have various wiring issues...hot and neutral reversed, ground and neutral tied, no ground at all, etc. Any of these can give your trailer fits.
A simple check with an "analyzer" (available at any hardware store:
Circuit Analyzer) will usually alert you to major problems in the pedestal even if it is only checking the 20amp plug, since, within the pedestal the grounds and neutrals are all tied together. Another way to test is with a volt-ohm meter, but, that requires that you know a little about what you are doing. For the really easy and, IMO, the best way to check is to get one of the various surge protectors (
Surge Protector) that are available. In addition to protecting your rig from low voltage (will ruin air conditioners and harm other sensitive electronics) and voltage spikes (will fry your electronics in a heart beat), most of them are also designed not to connect your rig to the pedestal if there is a problem with the wiring. They have a "testing" stage that they go through after you plug them in and before they actually connect the power. If there is problem, they won't allow power through to your trailer. I've always used one; a 30 amp with my 30 amp FunFinder and, now, a 50 amp with my 5th wheel. I won't plug in without it. I've had several instances with pedestals that open grounds and one that had ground/neutral wires reversed. Both alerted to me by the Surge Guard and I was able to move sites, and, in one instance, had the wiring fixed while I waited at another (it was a really nice site and they had a "repairman" on hand).
Usually, you can run all of the "installed" devices in your trailer at one time while hooked up. Most trailers are designed to handle the devices in the trailer (current wise) with usual usage in mind. Where some have run into problems is when they have the A/C running, something in the microwave cooking for breakfast and then they want to fire up a frying pan or DW has just gotten out of the shower while DH is scrambling eggs on the propane stove and she fires up the hair dryer. For instance, you should be able to have the A/C on, a snack heating in the microwave with the television and a lamp or two on with no problem. You get into trouble in the above scenario if you then add a toaster, hair dryer, frying pan, etc. into the equation. Any device that "heats" is a large load since they do it by resistive means; large amounts of electricity that produce heat from resistance in metal coils and a they can overload your trailer very quickly.
Edit... Yep, I would have some SERIOUS problems if I had two DWs...even though we've been married for 44+ years, I don't think wife #1 would take kindly to a wife #2; I know for a fact I couldn't handle a second wife, the first is more than a handful

Old age...second DW changed to me (DH).