Applying for jobs that require "Strong oral communication skills"

Applying for jobs that require "Strong oral communication skills" I am a covert stutterer and some days are pretty bad while other days I can get by without feeling too depressed. I am seeking employment right now and of course they all list "strong oral communication skills" is essential. Specifically, I am looking at a Claims Adjuster position at an insurance company which is local to where I live right now. Based on my education and experience, I believe I am qualified but I like most stutterers I am not confident in my communication skills. Also, I'm not entirely comfortable being a negotiator which I believe would a be a strong emphasis in this job. Does anyone here have any experience as a Claims Adjuster or any other job with heavy phone interaction with customers?