The rig should have a converter. When plugged into shore power the converter charges the batteries. The 12 volt items will operate on the converter.
When not connected to shore power the rig will operate off the batteries. You will not have any 120 volt AC available.
Your brake problem could be improperly adjusted brake shoes. As previously posted, you need to get manual for your brake controller and adjust it. Adjustment will vary for each tow vehicle. There could be issues with the wiring for the brakes on the trailer like worn brake shoes or magnets. I would start by adjusting the brake controller. Sometimes in a tight turn the brakes could lock up on the inside wheels of the turn.