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Hi my name is Robert....
Normally you would know if you are getting shore power if the outlets are working, the microwave and TV'S, they all run off of AC power when you are plugged in.
You would need a volt meter to check the battery voltage at the batteries, when you are plugged into power the inverter or power converter should charge the battery, when you plug in you should see about 13.5 volts and it should not go under 12 volts when you are plugged in.
IF you have more questions or concerns please feel free to contact me, IF you are satisfied with my efforts please feel free to select a rating at the top of the page, it is the only way I get credit for helping.... Robert