Hello Nancy, it sounds like you have instability in your CMC joint. I live in northern Wisconsin and had my procedure performed at the Marshfield Clinic. It is called a pyrolytic carbon implant of the CMC joint. It does look like a peg. There are numerous videos on Youtube for this procedure. The Mayo Clinic offers this procedure but my clinic was more convenient for me and I knew my surgeon already.
Well, I just had thumb #2 done 3 days ago. I have a pain pump and am in zero pain at least for one more day when the pump gets removed by my wife. O.T. is not going to solve your instability if you have biomechanical issues with cartilage and bone deficits. Xrays confirmed my specific problem and based on surrounding bone structures, I was a good candidate for this procedure. I am an active 59-year-old male who refused to settle for the barbaric LRTI method or fusion of the joint, although, those two options have been around forever.
My thumb was NOT in need of stabilization and NO tendons were used. I was offered a fusion of the joint or LRTI arthroplasty by the local orthopedists. I refused both of those options. The words joint replacement are probably misused. An implant is more appropriate and to the point. I would point out that the fusion option would stabilize the joint from dislocation, but the range of motion would be diminished(but it is probably already diminished from protective use).
Sacrifice is the key word here. I had to travel 150 miles one way to have this procedure performed by a specialist I personally researched and tracked down. There is limited specialization in northern Wisconsin so I had to travel. Getting a motel before and or after is almost a requirement. I, of course, needed a driver/caregiver for after surgery.

Best wishes moving forward.