Finally, we have some progress in restoring good brakes on the trailer.
A friend volunteered to help look for an electrical problem. To make a very long story short, we tried many things and could not come up with a significant problem in the wiring. We did add grounds.
I took the Airstream to a small local RV repair shop that was well recommended. They in turn looked for an electical problem to no avail. The first comment they made after inspecting the brake assemblies was that the brake drums appeared to have been replaced. I advised them that I was certain that those drums were original equipment. (I bought from the original owner, who in 4 1/2 years virtually never used the Airstream.) The brake assembly - backing plate, shoes, magnet, etc. were made by Dexter, which these folks consider to be the leader in the industry. The drums, however, were made by Reliable. While Reliable drums are OK, they said, they are not nearly as heavy as the comparable Dexter drum.
We decided to replace the entire brake assemblies and drums. We used an oversized one at that. There was some scarring of the magnet and drum surfaces, but no one that inspected them initially thought that they would not perform properly. However, the possibility of degraded magnets, magnet wires, and drum surfaces was the only good option left to us.
The good news is that the brakes are now working as expected.
The many folks on this forum were essential in narrowing down the possibilities. I thank all of you very much! Steve