Thanks for the input.
Just to close the loop -- first, I did measure the chain. It was stretched, which was why I replaced it. Second, I discovered (to my embarrassment) that the really serious problem I was having was caused by the fact that I had improperly remounted the rear wheel -- the QR skewer was not tight enough, and the wheel was slipping under load. That's what was causing the slippage in the larger cogs.
Finally, I do think this cassette is worn, though -- I'm going to replace it.