A stutter is an involuntary impairment in your speech mechanisms. A lot of people that stutter develop coping strategies. Losing eye contact is very common. Some will start tapping or rocking back and forth. Some will take a deep breath before talking. These are things that a stutterer will consciously or subconsciously start doing. If they do this enough it starts to become an automatic response when someone stutters. The path to fluency is to strip away these “struggle habits” and get back to the base stuttering symptoms. I don’t know you and I am not a speech therapist. I am just an expert in stuttering with more than 30 years of first hand experience haha. I would suggest that you are describing a coping mechanism that you adopted in the past and has become an automatic response. I suspect that if you record yourself or be really self aware when stuttering you will better understand what is going on. With a bunch of practice I suspect you could eliminate the hyperventilation part of your stuttering. Hope this helps. I wish you the best of luck in whatever comes next for you.