Many owners of that era discovered just what you discovered—they only had a black tank. Many have found ways to add a grey tank and fix or replace the black one. There must be threads on how to do it—try to the search function. Or look for threads on the restoration menus.
There are tank and valve cleaners available. You may have a blockage at the valve (usually old, hardened toilet paper or something someone dropped in the toilet) causing some of your problem. Check out cleaners at RV stores and try that first to see if it clears up the black tank problems. If that doesn't work, then it gets nasty cleaning the tank.
You may not want to do it yourself, but you should know how it is done so you can make sure you don't get overcharged or ripped off.
The Airstream is sold; a 2016 Nash 24M replaced it.