I believe a bad battery can cause the problems you describe so if you dont trust your battery, I'd start there. First, just remove it from they system. I'm fairly certain many 12v systems will work without a battery on shore power. If that doesnt work, with the rv battery still disconnected use jumper cables from the camper leads to your tow vehicle.
Also, I highly recommend getting a multimeter. It will help enormously when investigating electrical issues.
Hope that helps.
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