I agree with trying to lengthen your stirrups but riding without them will help great amounts towards finding your rhythm. I find that the majority of my students with their own horses come to me riding with their stirrups too short. That leaves them trying to balance in their feet & legs instead of in their body. Pay attention to where your legs fall when you are riding, are shoulder, hip & heel in line with each other? Try standing on the ground like you were sitting on your horse, try putting your legs ahead of your body (you will fall on your butt) then try putting your legs behind your body (you will fall on your nose) Try finding your exact center of gravity this way on the ground and then try to find it on your horse.
Have you ever seen the toys called weebles they used to make for small kids? They looked like a little egg shaped guy that you could smack around and it couldn't fall over. The reason they can't fall over is because they have a ball that rolls around inside creating counterbalance. Think of yourself having a ball inside from belly to seat bones. Sit in a chair or on your horse and practice moving your ball without moving your seat. Over exagerate at first rolling your stomach forward/backward, up/down, side to side. You will look & feel silly :) Eventually it will become very small virtually unnoticeable movement and you will be able to flow with any gait of any horse. Those that kind find this kind of movement are the riders who you will hear referred to as "can stick like a tick"
Hope this helps! Good luck!
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