I wonder if this horse was trained using that extreme side to side flexing thing that the western trainers do. I noticed on TBs especially, once you separate the head and neck from the shoulders, they are like riding a slinky. Luckily they're also very smart horses and easily retrained in most cases. I would work a lot with this horse at the walk just keeping his head attached to his shoulders. Even when you are doing bending/flexing exercises, make sure his head never breaks the plane of his shoulders. You'll essentially be teaching him to put himself back together.
Another good tool for riding slinky's is riding forward, forward, forward. The trick here is to provide contact he can reach into and the rider keeping a dead consistent rhythm that he can tune into and match. This works best trotting on a very large circle. The centrifugal effect of the circle puts him into the outside rein and helps him balance himself.
Hope this helps.