Instead of going the neuro route, I am going to say that he's just a very green horse. You said that he hasn't been ridden very much since he came off the track? Weaving back and forth across the road and "shouldering out" in the direction that they desire to go, such as back to the barn, gate, or toward other horses is pretty typical for a green horse to do. The best you can really do is to correct it as it comes up and continue to ride him as much as you can. For the shouldering out, getting him responsive to leg cues can solve a lot of your problem there.
As for the tripping, I would get his feet done first and then have someone watch as you ride him (or video you riding him). Some horses, especially those that have lived their entire lives on flat tracks or in flat paddocks, never learn how to pick up their feet and they end up as habitual trippers/stumblers. The only thing I've found to help them is to ride them a few thousand miles in country that requires they pay attention to where their feet are at...rough hills, deadfalls, things like that. Another part of that could be that he is still just incredibly unbalanced under saddle. The only solution for that is to make sure that you are balanced and then spend more time on him, asking him to use his body correctly.