I am a stutterer. I have struggled with it my whole life and I don't think you overcome it. You just learn to manage it.
I went to a speech therapist when I was about 8 and again in high school. They both told me that it was in my head. Stress (created in my head, apparently) made me tense up the muscless in my neck and formed a kind of blockage that made it difficult to speak. I agree with most of that - I can feel the muscles tense up and when I force them to relax the stuttering goes away.
When I feel secure and happy/relaxed, the stuttering is completely gone. When I am nervous, it's harder. Over the years, I have "practiced" enough that I am now a Trainer for a large corporation and travel around the country teaching classes. No one - including my boss - knew I had a stuttering problem until I told them.
Please go forward with the speech therapist - there are some physical and emotional conditions that cause stuttering that are completely treatable. Then there are those of us that have the problem "in our heads".
Even though I am mental, I have a perfectly normal life. I get embarrassed by it occasionally, but it hasn't been an issue to me, my family or my career.