One of my professors and eventually my mentor for my clinical fellowship stuttered. His name is Dr. Rodney Gabel and he is highly influential for many people who stutter. Dr. Gabel is a testament to show that you can do anything you want if you stutter. However, consider cost of education and potential wages. If you're looking at psychology, pay is going to be similar to a counselor or psychotherapist. Under $100k per year until about year 10 or so potentially. In most settings you may never make that much.