Montana, most horses are born with a side that is more supple than the other. A majority of them are born more supple or soft on the right side, so your mare fits into the norm. Part of our training process is to slowly practice exercises that enhance suppleness, with the goal of making our horses more even on both sides.
There are many exercises to practice; such as shoulder fore, shoulder in, haunches in, etc. But before you tackle those, it sounds like you need to do some basic homework to make her responsive laterally to you leg aids, since she cuts in, as you say, and ignores the holding effect of your left leg when you open your left rein. Without response laterally from your inside leg the above exercises are quite difficult, to say the least.
Since you say she responds better with no stirrup, my suggestion is to first lengthen your stirrups about 4 holes and teach basic leg yield to her. Start at the walk, and once she is responding well, go to the trot.
Get her really nicely reactive to your leg laterally on both sides and use a small dull spur, if necessary, as an aid to your leg. If you have control of your toe angle you can do this easily. Some riders’ conformation results in them having a naturally wide/open toe angle where the spur can easily be on inadvertently. Some mares are quite sensitive to the spur so no sharp edges. The spur I like for sensitive horses and more novice riders whose leg perhaps is not ready for a spur is called Spursuader. Here is the link for that: spursuader.com
The above exercises can be found on Equestriancoach.com in Fundamentals of Flatwork. The exercise I think you will benefit from to supple your horse’s left side is on Fundamentals of Flatwork Part 3: Advanced, chapter one, The Warm Up. She will have to respect your left leg for that to be effective, so do the basic leg yield first.
