Ah! Problem solved!
What had happened is that one of the pickup contacts had broken loose from the wire, and I had to remove the leads from their clips and resolder. Well, I had put the wires back into the clips with the polarity reversed. BZZZZZ! Once I switched the leads around the buzzing went away.
FYI, the buzzing would stop when I touched the jack, the shielding paint, or the pots. It was puzzling because I had tested all the grounds with my meter and they were all showing good contact.