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Hi my name is Robert.....
IF you replaced the battery it sounds like a power converter or breaker issue. Sometimes they will stick an auto breaker on the cable from the converter to the battery, below is a picture. Some are also thermal, they heat up and break contact (lights go dim) then they cool off and reset and the lights would go bright. As a general rule if you had one it would be 18" from the battery and could be on the frame. To test you would check voltage on each post to ground, the voltage should be the same or close. You also might have more then one, sometimes the landing gear or slide-room will have one.
The other issue is the power converter, the power converter will take the incoming AC voltage and make the DC voltage. The converter will be part of the main fuse/breaker panel or stand alone and plug into an outlet.
One other item would be if you are using an adapters or extension cords sometimes they can fail, if using an extension cord you can have line loss when you start turning everything on. IF you are using any cords or adapters I would plug in without them and see if the problem goes away...
Hope this helps some... Robert