Hello from Colorado: Welcome to the vintage Airstream hobby. We have a 86 Limited 34'. It is likely constructed similar to yours. I have done considerable work on this trailer, as well as a 66 Trade Wind 24' and now a 75 Overlander 27'. Vintage Airstreams is a hobby of mine.
First I will offer my opinion on the Airstream trailers I so enjoy. They are built somewhat aerodynamic and light weight to tow more easily. They are designed efficiently for the smaller interior space. And they have an all aluminum body that lasts a very long time. That said, aluminum is an excellent conductor of heat and cold. The Airstream walls are only about 1 1/2" thick with ordinary insulation installed. They are more difficult to keep warm in the winter and cool in the summer. Many Airstreams leak rain water that tends to rot the wood subfloor and thus weaken the trailer. Airstream appliances are ordinary RV fare and tend to wear out or be problematic. I don't consider an Airstream an ideal full time coach. I think big fifth wheels and motorhomes make better full time units. There are many folks that disagree with me and live full time in their Airstreams.
I think mission one is to assess your Airstream from the bottom up. Inspect everything and make a list of what doesn't work. Tires, brakes, bearings, axles, belly pan, doors and locks, windows and seals, all appliances, all plumbing, all electrical both
12v and 120v. You have already found a leak from the fresh water tank area, and maybe a leaky window. A thorough inspection with a list of problems noted will allow you to develop your project plan. Maybe you did this when you bought the thing.
Then start knocking them off one at a time.
My "fresh water tank" was leaking in the 86 Limited. A leak from that area isn't necessarily a leak in the tank. They have a drain valve that is infamous for leaking. My leak was due to the plumbing above the tank. I curse Airstream for running plumbing above the fresh water tank but below the subfloor. I think our trailers were plumbed with poly butyl gray water lines. I had one fail and leak. So I dropped the tank and re-plumbed the lines with PEX, a better material. See photo...
Hope this helps. These Forums are an excellent place to learn about all things Airstream.
David