Sounds like you have run into the WONDERFUL PONY ATTITUDE! No matter what every pony has that stubborn attitude. I train alot of ponies and most of them come to us with the problem of taking the bit and running, and also not being patient. This is the middle stage of the "pony attitude" for me ha ha ha. It sounds like your pony is right there.
Normally your pony is probably loved on and spoiled (i spoil my girls too). She/He is probably used to getting what they want. Now they are trying to push their limits. You need to stay firm. When you pony starts to walk away when you get on stop what your doing. Grap the rein across from your (right side) make the head turn that way and make them stand. This confuses them because you are on the left side, but yet something is pulling them from the right. Then mount agian keep doing this till they stand still. Once you get on give them a big pat for standing sitll even though it took you probably a long time to get on.
For the wonderful taking the bit and running part. Its not as simple. It sounds like your pony may have a hard mouth if you have gone through bits. First I want to ask if you ride with a standing or running martingale. If you do keep it on. If you don't I would reccomend trying one. This gives you more control of the pony's head. It sounds like a full cheeck would help you with the turning but it may not be enough bit. Have you ever tried a slow twist full cheeck. This is not a horrible bit becuase it has a twist, it just aids in the help of the pony listening. Also the full cheeck makes the pony turn its whole head to turn so this will help with turning. If your pony puts her head down when she trys to keep going I would recommend a 3 ring/elevator bit. This has 3 rings (like the name) the bottom is the harshest the middle is the in between and the top is just like a snaffle. This should help you in stopping and probably turning.
So in the end I would try a 3 ring and see what happens.
Sorry for such a long response, but I know alot about what you are dealing with.
Let us know what happens
and good luck