It can never hurt to have his teeth checked out (should really be checked by a vet/dentist because they can then correct any possible issues he has), but I am much more inclined to believe that this is a training issue. Because he is still hard in his mouth, he was likely never really properly
re-trained after retirement from the track.
Truthfully, I would strongly suggest that you find a place where you can either work with a trainer or take some lessons on your horse. Not only will that help you with the problems you are having now, but it will make you a better and more confident rider in the long run.
With lessons, you wouldn't have to do like many others did and learn all your lessons the hard way