This is exactly what good stuttering therapy can help with. You're correct in identifying the reason for your trouble with eye contact: shame. I don't know how long this has been a problem but it might have become so ingrained that eye contact really does feel completely beyond any control. I wonder if there is any situation at all where you stutter and are able to keep eye contact. It can be a parent, best friend, even a pet. Are you able to do it if you look in the mirror if you stutter in such a situation? I think of eye contact as like a foundation avoidance behavior. When you don't have eye contact, you don't have a platform upon from which to build which makes everything else more difficult. I would seek out an SLP trained in stuttering who you feel can help you with eye contact. These things are hard to work on on your own and though it's not easy to find one, the search for a good SLP is well worth it.