So, I had to leave this project for a bit (summer vacation with the kids), but I got back to it.
I checked the shaft, and it looked fine. I then put together a makeshift flapwheel out of a dremel sander and a bit of 1000-grit sandpaper, and used that to clean up the inside of the gear (very lightly and carefully!). I reassembled it, oiled it up, and I am happy to report that it is now working great.