Alto is great fun for me too. I think many tenor players like to dabble in it because they really respect and like all the widely varied alto tones and styles of the greats, like Earl Bostic, Sanborn, Desmond, etc., and a lot of great alto styles just do not translate to tenor that well, and vice-versa.
Greene, for the thumb, you might try this simple exercise - do it on both hands, and its something you can do anytime when not using your hands - just touch the fingertips to the thumb in any order. You may find the the right hand thumb and little finger hard to do at first but this simple exercise will help with saxophone thumb hook pain. Beyond that, you may have to take some kind of action to fix what is causing this 'use injury'. I recently removed an extra-long brass thumb hook from my VI tenor and put a plastic one back on. I had a little collection of black hooks and could not decide which one was the original so I picked the one that was longest (but not as long as the brass one). After a gig, I did notice improvement in how my right hand felt. You might also need to take a fresh look at your strap strategy as it affects hand position and perceived weight on the thumb. Hanging the sax low will cause thumb stress as well as adversely affecting your neck and sax mouthpiece height/angle. Maybe we don't want to go 'Sanborn' with the sax really pulled up high but there is probably an improvement there for anyone. We all just get used to doing everything the same way, which is okay until it makes something hurt.