I had a TB with a terrible rushing problem. My first trainer wouldn’t even work with him, but with a new trainer and some time, he ended up eventing successfully through preliminary.
Although he had been jumping for a few months before I got him, I took him back to ground poles and cavaletti work. First at the trot, then at the canter. Do them a bazillion times if it takes that long until its no big deal and there’s no rushing.
Once we had the ground poles down, we started doing a lot of gymnastic work. Bounce-1 stride-bounce. That type of thing. It helps the horse learn how to balance himself. The trick is to just stay put once you’re in the gymnastic. No helping the horse find distances, place his feet, etc. Gymnastics should help a horse find his feet and balance all on his own.
Lastly, learn that circles are your friend. Once we were jumping in a controlled manner, there’d still be a bobble here and there. I would circle a few strides from the jump if I felt him start to think about rushing. Make sure you don’t leave the circle until you have a balanced canter. Winging around in a fast, crooked circle won’t help anything. Once you’re happy with the canter, start your approach again. If he rushes again, circle again, closer and closer to the jump if need be until he gives you a polite canter to the base.
Rushing isn’t easy but can definitely be managed with enough patience. Good luck!