Definitely sounds like some lunge lessons would do you good until you feel like you have a balanced seat. When you have a solid seat, if a horse does do a sideways jig, you'll stay on. It can be scary riding a horse. We know what can happen to us, especially you with your broken arm... OUCH!!
Since this is something you want badly, just take your time with it. Take it nice and slow. Our trainer always says "Rome wasn't built in a day." Since you already have horses and you enjoy them so much, don't worry about the fact that you can't ride them right now. See it as a goal you are working toward in your future. Don't put so much pressure on yourself.
Also, do you have a dead broke horse you can learn on? If not, get one. You need a horse that you can feel comfortable on. Of course, a horse is a horse, and things can still happen. But if you have a dead broke horse, that brings the risk factor way down.
Good luck and keep us posted!!