Welcome to the forums and congratulations on picking up a great trailer.
You're right that there is only one tank -- the black water tank -- for holding toilet waste. The grey water -- shower and sink water -- used to be dumped right to the ground. Back in the early '60s, they would dig a "gopher hole" and let the water run right into it.
That's illegal today, but you can get a Blue Boy
that will allow you to collect grey water and properly dump it. Eventually, you may want to install a grey tank within the frame. Frankly, I'd spend some time getting to know and enjoy Airstreaming before getting too carried away with renovations. The motivation will help tremendously.
You will find this forum to be a gold mine of Airstream knowledge. The search function will bring up quite a few threads on Overlanders and will hopefully answer a lot of your questions. If not, just post here. You'll find the Overlander group to be a great resource.