Well, this will take some sorting out first. Attached is a wiring diagram from my 66 Trade Wind It may help you as Airstreams were built rather alike. However, 1966
was a significant model change, so I don't know.
Airstreams are wired to run on 12v and propane. At the time they called it "Univolt". I think Airstream was a pioneer in this type of RV electrical systems that is still used to day for the most part.
So the "heart" of your electrical system is a "converter". Yep, it converts 115v AC to 12v DC. Your water pump, vent fans, fridge, furnace fan, etc. run on 12v. Your trailer works on 12v while plugged in to 115v because your converter is working. Good! Your new 115v water heater won't work, and if you have an AC on the roof, it won't work either while on 12v.
My 66 trailer has a battery, and the red wire is on the + and the white wire is on the -. It appears you have identified these two wires. They ought to be in the 12ga or 10ga wire size. Pretty beefy wires.
There is a 12v fuse box in your trailer somewhere back there. There will be three 12v fuses protecting the 12v circuits from overload (like a failed fan motor.) You need to make sure those fuses are good.
So I guess I would unplug the 115v and hook up a 12v battery to your red wire (connect first) and white ground wire and look for smoke! If none, then
start trying out your 12v lights, fans, furnace, etc.