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Hi my name is Robert....
Sounds like you are missing all the DC power when you unplug. When you are plugged in the power converter will take the incoming AC power and make the DC power and charge the battery. When you unplug the converter will switch to battery power. So this sounds like the converter is at fault OR a tripped breaker near the battery. Sometimes on the cable from the converter to the battery you will find an auto breaker, below is a picture, most have a small tab that pops out when they trip. As a general rule they like to put them 18" from the battery, the voltage to chassis ground should be the same on each post, I would take a look to see if you have any. IF you do not see any or they test good then I suspect the converter is at fault, If you have more questions or concerns please feel free to contact me... Robert
This could be an issue with the power converter, when you unplug the power converter it will switch to battery power, sounds like it is not switching over. When you plug in the converter makes the DC voltage. Do you know the make and model of the converter?