Adam, this is EXACTLY what happens. You see, you have taught your brain (from years of repetition) that saying your name = stuttering = humiliation and embarrassment.
So now what do you have to do? Prove to your brain the exact opposite ie. saying your name = fluent speech = calm = not even an important event.
How do you do this? By continually using a crutch and showing your brain that saying your name is not actually something to fear. Your mind will eventually wash out the stuttering memories with fluency memories and it will completely forget to fear saying your name because you have given it no reason to continue to do so.
I cannot remember the last time I even thought about saying my name. You will get there too, just have some patience and trust that it will only take a couple of weeks for this new fluency habit while saying your name to form.