The old chain was stretched but not wildly so -- the chain checker measured 0.5% but not 0.75%. The chain was really dirty though, which probably accelerated the cog wear.
I just realized that a new chain is going to really exacerbate the skipping problem. The increased pitch of the old stretched chain would make it easier for the rollers to make it past the hooks on the teeth and lay into the sprocket. I'll get some pictures of the sprocket and post them here, I think the teeth are hooked but it's really subtle and is hard to see visually.