I do not drive horses (I have once or twice, but really holding the reins(?) doesn't count). However, it seems to me that you have only 3.5 options.
1. Sell the horse and get a new one
2. Sell the cart and get a new one
3. Hook the horse up to the cart and go.
3.5 Modify the cart to make you feel safer and then hook him up and go.
Sounds like you either have to face your fear by DOING or else never drive this horse/cart combo again. Perhaps having your daughter take him for a jaunt in the cart first, or riding with her if it is possible will help your fears.
As for all the opinions on the OP's first post. We've all done things for which we are ashamed; we've all done things that others would disagree with severely. I certainly cannot comment on the methodology of driving, but OP, don't get caught up in what you did in the past. You can't change it, you can reflect on it but don't get worked up. It is done and over, and you have to decide to move toward the future.