OP, giving him support with shoes or whatever won't help if his feet aren't balanced.
Just as an example. I brought in a customer horse a few months ago that moved similar to that when he arrived, though not quite as bad. Because his feet were overgrown anyway, I had my farrier out just a few days later and he pointed out to me that the left heel on every single foot was taller than the right heel. He trimmed the horse balanced and had me walk him out before putting shoes on. The swivel was completely gone.
I'd check your horse's feet yourself. What I've noticed, if farriers are doing something wrong, they sometimes just don't know that it's wrong. Whether it's the way they look at the foot that makes it "look" level to them, I don't know.