I will mention that part of the shifting and swinging leg issue may
be due to how the horse moves, though I likely couldn't say for sure unless I rode him myself or watched you ride him. I have one horse that does that to me regardless of how I try to sit his lope, he always flings me off to the right and makes my left leg slip way back. It is a genuine workout to keep proper position on him and it is done simply by pure strength
. He's the only horse I've ever ridden (in 25 years of riding countless horses) that does this to me. Do other folks have the same problem with this horse or is it just you?
As for sitting on the pockets, that just takes time and practice to be able to do efficiently. Can you control his speed at the walk and trot by doing the same thing? It's easier to start slower and work on your own motion until you've got it down before trying anything faster. If you can't slow his trot by sitting deeper, then you aren't going to have much luck at the canter either.
I don't know how you're currently riding, but regardless of what speed the horse is going, try to avoid just using constant pressure on the bit to keep his speed under control. Have him go on a loose rein and if he starts speeding up, then only pick up the reins until he's going that slow speed again, then let the reins go loose. If he gets really bad about it, every time he starts to speed up, have him break down to a trot or a walk or, even better, stop him and back him up a couple of steps before going on.
As for the bit, well, I would change it in a heartbeat if he was mine, but you likely don't have any control over what bit you use, do you?