I'm not sure, but I think I have a similar problem. One thing: check your seat. Is one leg usually a little ahead of the other? I found I was riding skewed, not a lot, but enough that one rein was always a little shorter than the other.
Second, work on developing the right rein contact, drop the left. If your horse goes right, fine, for a while just keep the contact soft but honest. I'm right-handed, so I had a problem of my horse wanting to fall left. I began to do more flexing that way, soft and slow, and it actually only took a few days before she started stretching into the contact. Use only 1 rein for awhile, so as not to confuse your horse.
Riding plenty of different-sized circles, changing direction a lot, and mixing in with leg-yields, helps to develop balance and symmetry.
Also, if you don't, I recommend your always wear gloves. It helps stablilize the hands. Hope this helps!