I can not even begin to imagine how difficult that situation was for you. I don't really understand it, but I think any time you have trouble communicating things begin to be frustrating. Being new did not help matters.
Firstly, I think the other poster is correct, you need to seek out, or continue whatever therapy you can do for the stutter, it can get better. I am sure you know of or have heard of over the years the many brilliant people who had a stutter. You are not stupid and can be a good nurse...One step in getting there will be gaining better control over the stuttering. Patients of course will not be expected to be understanding of accomodating because they are ill and very concerned about why they feel bad, what the diagnosis means for them etc... BUT the other staff- SHAME on them for what was said.
Did you share with your manager/charge RN or preceptor that you have a stutter? I know for me, when I have some sort of issues, I find I am more confident when I simply share up front in a non-pressure situation. Something like, "I do have a stutter and it can get worse if I feel nervous. I've been working on it and it is getting better each day, but please understand that sometimes I might have some trouble." Thus, your charges etc, know whats going on, and can try to help you out, by possibly starting an admission while you get comfortable with the pt or what not.
You have a BIG challenge to overcome. But, you can do it. You made it through Nursing school
- which involved a lot of pt interaction etc. take some time to get things in place to allow yourself to succeed.