Practice. More practice. When you're done practicing, practice!!! ;)
When you do it WITH stirrups, don't let your leg fly forward and brace off that. What helps me is just letting everything relax and loosen up. I may be weird, but that's how I do it. Don't slow him down too much. It depends on the show you're going to about whether or not you need to slow down. For the schooling shows in my area I only slow it down just a TAD to keep it looking graceful because my pony's a tornado.
Keep trying it WITH and WITHOUT stirrups and take a day just to identify your problem areas and how each method is similar and different. Try to keep the exact position with stirrups as you do without.
When I school dressage on my other pony, the spring-loaded one (XD) I lean back a little because it hurts my back too much to stay straight up. But just a little.
Again, just practice. Don't sleep, just practice!
As for under saddle and equitation:
Under saddle(U/S) is the flat portion of a Hunter or Equitation(eq) division. Normally for the U/S it is a walk/trot/canter class and may have sitting trot, but most don't. Hunter U/S is judged on hunters (manners, performance, & way of going) and not jumping. In eq, it may have sitting trot and no stirrup work depending on the level of show and the division. Medal equitations usually have the flat with an extra test, or work-off. That can include jumping a shortened course without stirrups, a quiz, anything.
Equitation can be western, english, hunt seat, and dressage-seat. Not sure about western or dressage eq, but english and hunt seat eq is judged on hands, seat, and control of mount. Your horse should reflect your riding. You don't want to go around riding a horse that wiggles everywhere and bounces off the walls. Most eq classes do jump courses, which is hunter equitation. Some open shows have it on the flat and a seperate division for jumping equitation.
Hope I didn't confuse you! Good luck!