You have found the classic problem with training a horse to drive, and one that is too often not taken seriously enough. In order to be a god driving horse, it first must be a good standing horse.
We begin our training with training the horse to stand quietly while tied, whether in the cart or not, and always do the same thing when we put to, so that standing becomes an accepted part or the horses life. We often have to put to with only one person, so it is imperative that the horse stand quietly until we are in the seat. We always tie the horse while putting to, and it is important to tie to a gate, wall or hitching rail, something that is wide enough that the horse won't try to go around it when it is untied. (As you found out, a tree doesn't work) to start with, harness the horse and tie him and then just let him stand there for a while. Let him get used to the idea that having the harness on doesn't necessarily mean that he's going somewhere right away. Then when you do put the cart on, let him stand for a while again. Obviously you stay near by in case of problems (and there may well be some, at least at first) but let him get used to the idea. When you eventually feel he's settled enough for you to get into the cart, you can untie him, but have the lines in your hands so you have some control. If he's gotten used to the idea that he can't go ahead, he should just stand as quietly as if he is still tied. You can then get in the cart, gather your lines so you have control, but still have him stand. When you feel you, and he, are ready to drive, ask him to back a few steps so you can turn him and drive forward away from the gate or rail. The backing and turning before going forward are a key part of the technique, so the horse learns that he will not simply go straight ahead when he starts out, but will back, turn, and then go forward.
This won't be an easy thing to teach, because he has already got the bad habit of going forward as soon as he is in the cart, and you will need to go very slowly and repeat many times for him to learn the new way of doing things. Horses learn by repetition and eventually he'll get the idea .We always put to with our horses this way, and it does work. It will probably take a while to get him used to the new way of doing things, and you should have an assistant helping until he's good with it, but in time it will come to be second nature for him.
You didn't say how he does in harness, if he will stop and stand when you're on the seat and you just want to stop and rest or in the line-up at a show. You'll need dto stop him in the ring and let him just stand for a while. Pick a place with a nice view, and take a lunch or a six-pack with you as you might be sitting for a while. This again, will just take a lot of repetition. but eventually it will all come together and you both.will have a safe and much more enjoyable driving experience. GOOD LUCK!