A few things come to mind. Chiropractic and yoga to work on your straightness and balance. Find a chirporactor who specializes in sport. Feldenkrais might help. Do you stretch before riding? If you have one leg that is stronger than the other, or a hip joint that’s tighter than the other, it can pull you to that side.
I have had similar problems over the years. Have your instructor/coach, or just a friend on the ground, check your position in the saddle. Do you sit straight and balanced in the saddle as viewed from behind? Your helper should observe you in motion, not just at the halt. Do you collapse your torso on one side or the other? Do you contract your (right) leg up – even if it’s just a little? Pay close attention to the angle of your thighs relevant to your torso. Does one leg look straighter and the other more angled at the hip?
An exercise I have used in the past was to “play” with seatbone positions while driving. Especially at stoplights. I’d push one seatbone forward, then the other. Or I’d weight one more, and then the other. Since your horse is very sensitive to seat aids, I’d be careful practicing this in the saddle, though it’s fun to incorporate “seat only” exercises in warm-up. I did this with a student yesterday. Pretend you have no legs and use only your seat bones to turn your horse. Think of bringing your inside shoulder slightly back on turns and Voila! you weight the inside seat bone.
I will be watching along with you for other’s suggestions!