Answering the phone is a nightmare. Several times I've been stuttering like a fool as the other person says "hello" over and over. A ten second conversation may take a minute and leave me embarrassed.
I have a few stutterers in my life. They don't like talking about it, but it can't be easy. I get the impression that the pressure of having to talk makes the stuttering itself worse. Have you ever considered treatment? I know it never fully goes away, but there are ways of making it a lot less pronounced.
hi mitral In your situation I would have a recorded message to play - like an answering phone. You could play it when you pick up and explain you have a stutter. You need some serious help. There are plenty of skilled people who could help you. Best wishes :)