External frost-free facuet gradually slowing flow
After pulling everything apart a few times, I found it out. Looking at the diagram on the site, part 10 (the retainer screw) had loosened off but not to the point where it were no longer connected. If it had loosened off completely, turning the handle would have done nothing.

Taking everything apart, part 9 had been squished a bit and was stuck in the tube. Loosening everything off let me knock the retainer and valve out of the tube. With the tube cleared, I tightened the retainer screw and the valve against the valve stem assembly. With the retainer screw now attached, I reinserted the valve stem assembly and put it all back together. Great success, eventually.