Thanks to all for your help!
Update time: I dove in after getting home from work tonight. Getting it fixed was remarkably easier than I'd anticipated. The sleeve is indeed a push-pull, and not threaded, for what it's worth.
The two hex screws that thread into the mill's body come right out. I locked the quill and popped free the center screw where the quill handle/wheel is. It took a little bit of persuading once the handle and keys came out, but I managed to back the sleeve off of the shaft. I then took some brake cleaner, a brass brush, a paper towel, and then a bit of 600 grit to clean both the sleeve and the shaft. Reassembled easily, and I now have fine-feed back.
Again, my gratitude goes out to everyone who helped!