Pittsburghracer had it right: the culprit was a bad float (not the black plastic kind, but the old-style brass type). As I said, I haven't checked the float level in years and the old bowl had a sight plug, rather than a sight window, so who knows how long the float's been bad.
Thanks for the help! up

"When I'm in a slump, I comfort myself by saying if I believe in dinosaurs, then somewhere, they must be believing in me. And if they believe in me, then I can believe in me." - Mookie Wilson