Your horse doesn't sound like a good prospect for Driving. She is already not obedient at ridden skills, and Driving takes a LOT more patience, acceptance from the animal. Some are just not suitable, no matter how hard you train them, they don't have the correct temperment.
Here is my response to the other TB and Cart thread. Pretty much everything I posted would apply to your horse that won't stand still and has other problems. OTTB with a cart?
TBs usually just are not Driving horse material because of their temperment. TBs can do other jobs well, but Driving requires MORE SKILLS of the horse than riding ever does. Many folks try driving a lot of kinds of horses, but those horses wash out of Driving training because they don't have the needed mindset to succeed. Not worth getting horse or person hurt by forcing them to drive ANYWAY.