Loveland,
You were smart enough to buy a known name, so contact Airstream and see if you can get an owner's manual.
I have never been called to work on an Airstream, but if it is all typical, this will help.
One battery is the main engine and the other is the coach or house battery. The main engine should charge both. In most coaches, the shore power charger/converter will charge the house battery, but not the main engine (chassis) battery.
The genset is not likely to be LP fueled but pulls from your main fuel tank.
Matt