I agree with everything that everyone has already said, and want to add to your comment about strong verbal communication. I am just starting my career as a teacher, and as a person who stutters, this is something I thought I would never do, as it is basically like public speaking the whole day. I have learned that if you show confidence, you are upfront with people (adults and students, in my case) and you treat your stutter as something that makes you better and more resilient every day, your potential employers and future employers will not care at all! Parent-teacher interviews are the worst part of my job (as a stutterer) and this year, they were on the phone (my worst nightmare!) but being upfront about my stutter makes everyone more comfortable (me included!) and more patient. In addition, I always say in cover letters that my stutter has never affected my professional life (which it hasn't, even though I have gone home utterly exhausted from communicating most days). I find this gives the interviews a heads up as well!