It seems like you are going through testing the campground outlet the hard way. Just buy a 30 amp male to 15 amp female adapter and get a voltage meter and a polarity/ground tester both with 15 amp plugs. Plug the 30 amp end into the electric box at the campground and plug each of these into the 15 amp socket on the adapter. You will immediately test the polarity and ground and you will know the voltage coming through the outlet at that moment. Which would be nice if the voltage did not vary - but it can for a number of reasons which is why you want the Surge Guard or Progressive Industries Surge/Voltage Protector.
This will not only test what you just did when you first plug it in, but while it is connected between your RV or trailer and the campground box it will keep monitoring the voltage and should that voltage drop below acceptable limits or increase above acceptable limits it will cut off power to your electric system immediately saving frying your wiring and electronics inside. This is not uncommon to occur. It also provides surge/spike protection and will blow in the event of a spike or surge from a storm, etc. cutting off the connection.
Is it worth a repair and replacement bill in the thousands to save $250 to $300 to buy a Surge Guard or Progressive Industries unit? That is a decision only you can make. I never hook up without it.