Perhaps you could satisfy your interest in psychiatry by becoming a psychiatric nurse. I don't know if or how you could combine dentistry and psychiatric nursing in the same practice, and you certainly wouldn't have the same power and autonomy as a psychiatrist would have. But it would cost less money and take less time than going to med school and completing a residency in psychiatry.
There's also the possibility of getting a master's degree in psychology or social work, and becoming a psychotherapist through that route.
No matter what path you take, you'll take on a lot of debt, you'll be in training (schools, residencies, internship, etc.) for many years, and you probably won't advance as far in either field as you would if you focused your energy on one field. If it is what you want to do, it may work out. I wouldn't recommend it.
