Good news! I was on a different horse today, and things went sooo much better. Also, we did trotting on the lunge line so I didn't have quite so much to think about and do at once.
This guy was more responsive to me than the other lesson horse was, which is such s confidence booster, and his smaller body makes it easier for me to squeeze. He is in between the bigger horse and the pony, probably a 15+ hands. And while his trot is bigger than the pony's, it is not so powerful that it takes my breath away. It stretched my comfort zone without going completely outside of it. I did not post perfectly, and I'm sure it wasn't pretty while I was sitting either, but I was able to keep my hands down with occasional reminder, to think about putting my heels down, and and to remind myself to breathe and try to relax. Basically, I wasn't in a blind panic.; )
She said we will see if I can keep riding this guy. He is a better fit for me, overall, both physically and mentally. Though the other horse is very good and mild tempered, I was always intimidated because of his size (he's a beautiful warmblood with a nice strong round behind), even though that wasn't necessarily logical.
Feeling much better and encouraged. Now I can actually focus on all your good tips. : )