We practiced the saxophone again last night, and I love that time with my son. And I think I figured out something important! I think I get a better sound when I use a little more of the mouthpiece than I had been. It also helps with that issue I was having when transitioning from high notes to low notes. It is not an issue going from G to high D, but going back I would always have issues on the return to G. I would either be an octave too high (without using the register key) or I would squeak.
With more of the mouthpiece, that seems like it's working a lot better.
I had a really long day yesterday. I didn't even leave work until after 7 (my usual time is 4:30). I thought I was too tired to practice, and he was reluctant too, but we did it anyway. I am really glad we did, because we both had a good time. My son started out lethargic and a little put out about having his very important Hyrule Warriors game interrupted, but by the end of our session he was enthusiastic. And later in the evening he was downright bubbly.
I am really glad I made what I thought was an irresponsible purchase at the time. I am really enjoying playing an instrument again.