Thanks SeanB and david77 - that was helpful!
I turned the pot to the bad spot with the machine on, then turned it one more revolution. I then took it apart. Turned out to be super easy - the case was made up of three parts - the front that also had the shaft fairly permanently attached to it, the middle which was mostly just a cylinder, and the back. The front piece had two metal clips that kept the back pressed against the middle and the middle pressed against the front. A screwdriver was all that was needed to pop it the clips off of the back.
The insides looked pretty good except for the wires looking a bit corroded. I was very impressed with how uniformly the teeeeeny wires were wound. After I opened it I cleaned the area one turn above the wiper with deoxit and then with alcohol. This fixed the problem area, but then I realized there were a couple other more minor problem areas. I gave up and gave the whole thing a wipe down with deoxit, then cleaned with IPA, and then put a little deoxit gold on it. Closed it up and it is working beautifully now.