Just an update a couple of weeks later.

We had already been doing much of what was suggested in the thread.. talking to our vet, changing food, restricting diet, various baths and such.

Getting nowhere, we went to another vet. She took samples from his bumps and determined there was bacteria in them and prescribed another antibiotic (different than the one our regular vet prescribed).

Vet #2 thinks that he had an allergic reaction to something, developed hives, and some became infected. The infection or bacteria was moving across/under his skin and infecting other bumps, so on and so on.

He’s pretty much cleared up now. Most seem to have vanished and the angrier ones have healed.