Yes, there are plenty of people who will tell you that the part draft is why she is heavy on the forehand. I have had my Clyde cross for 17 yrs. He is extremely heavy on the forehand and always has been. However, I have recently learned ways to make him lighter...I just don;t have the motovation to try them,since he is basically retired. Anyway, when we were still jumping he would get SOOOO heavy. I finally switched to a Waterford. Now I have no idea any more what is legal and what is not, nor do I know what a "bubble bit" is. The waterford is such that they really cannot lean on it as well as a more solid bit. I have one that is a D ring, and I use a gag for cross country. It helps, but does not solve the issue, as others have said.
My new guy came to me riding in a Tom Thumb, and was a bit strong at times. I do not jump anymore, and have gone to Western, but have spent many many months of trainers and hours of riding getting him soft. Totally soft. A couple of things specifically helped me. I learned that as soon as he pulls, if you pull back, they just brace harder. Drop them like a rock. Obviously, you should not try this cross country, but in a ring. THey start to pull-let go. THey will learn that pulling does no good. I frankly got away from a bit altogehter for about 6 months. Rode in a rope halter and found he was much more responsive. You may find that a bitless is actually better. I had one horse afraid to go forward at all in a bitless.
When your horse speeds up, don't fight and pull-do circles. When you get the speed you want-go back out on the rail.....when they speed up-circle again. Serpentines are your friend. Concentrate on using your body to turn your horse and NOT pulling! As I said, I have spent hours-many just walking (which is hardest, actually, like riding a bike slowly)using NO reins at all, turning the horse with (exagerated at first) body and leg cues. I have also come to have a much better understanding of using your body and weight to get your horse slower (as well as faster), and have mine trained to a very quiet "shhh" to slow (draft does this really well, actually). Sit very deep, breathe out and "shhhhh".
Good luck-maybe NO bit will be your solution!