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Lights, refrigerator and furnace all operate on 12 volts, so you can use them without having to plug into 120 volt power. 12 volts comes from the trailer battery which is charged by the converter when you're plugged into power.
When you say your lights looked like they were burning out, do you mean they were getting dim? If so, it sounds like the converter isn't working and you've drained the battery.
Make sure you have 120 volt power getting into the trailer. Do you have something you can plug into a 120 volt outlet in the trailer to verify you have incoming power? Also check the connections on the batteries, make sure they aren't loose or battery acid hasn't caused corrosion to build up on them. Beyond that you'll need a voltmeter to track down where the 12 volts has gone missing.