Oops...didn't see the Tips for Balance thread before I posted. Sorry for the.duplication.
I'd suggest doing a lot of stirupless work. It's very easy & common when learning to put a good amount of pressure into the stirrups and essentially brace for balance. It is however, counterproductive and will unseat a rider.
Balance comes from having an independent core & seat. Think of how a belly dancer moves, hips and midsection can flow independently without effecting legs or upper body, stomach can "roll" up and down. We as riders need to have similar independent movement. Think of your core as your "shocks", the concussion of the horse's movement should be absorbed through your middle.
I explain it to my little riders with weeble toys as an example. Silly but it is a good visual. They work because of the ball bearing inside that rolls around as a counterbalance. Think of having that ball bearing in your middle and "roll" your ball with your horse's movement.
See if you can do a lunge line lesson stirupless. Start at a walk and over exaggerate the movement through your mid section. Try to move with and count the footfalls of your horse. Then do the same at a trot/jog. If you start to lose your balance, grab the horn, take a break, try again. It takes a long time to find that true feel & independent movement.
I also have students hula hoop. Sounds silly but it helps loosen up hips and get a feel for the "wiggle" a rider needs in their middle.
Do "airplanes" with arms straight out and make circles in the air, put one hand on top of your head/switch, while stopped take your right hand to left boot top (w/out stirrups) lh to right boot top focusing on keeping your seat in the saddle, doing some of those types of exercises will take a bit of focus from "riding" helping you relax and sort of fall into it.
Hope that helps!