Here's what I get up to on a regular basis from a communications standpoint, in order from most to least common:
I write a couple of thousand words worth of emails and other communications each day. This is by far my most common method of communication, especially as I work in a large organization. The content ranges from the usual setting up meetings/making requests for information to making more long-winded explanations of designs or UX concepts to my other business partners. I may be preparing a presentation of a new mockup, which also takes up quite a bit of time.
Internal Spoken Communication
I don't lead many meetings in my organization, though I participate in quite a few. My usual in-house spoken communication is making comments, suggestions, or asking questions in our internal meetings.
The meetings I do run usually involve me presenting a new design mockup to various team members, presengin the results of my most recent UX tests to my business partners, or running whiteboarding sessions with internal stakeholders.
External Written Communications
I'll occasionally need to write e-mails to our clients asking them to participate in UX studies, give their feedback on a new feature or service, or gather requirements.
External Spoken Communication
I occasionally run UX sessions that require over-the-phone or in-person communication with our clients. This is probably less than 5% of my regular work, but I do this on a monthly basis.
I wanted to take a moment to share some of my wisdom that I've gained through years spent working towards professional success while overcoming a number of my own personal stumbling blocks.
Somewhat like you, I struggle with anxiety and some form of speech impediment. Both of my blockers are a little different, however. My anxiety is generalized, and though it often manifests itself in social situations, it affects every aspect of my life. Rather than a stutter, I have some difficulties with dyslexia, which translates to speaking words out of turn or occasional difficulty understanding spoken sentences.
For me, my anxiety was a much worse impediment than my dyslexia. My anxiety was, at one point, capable of negatively affecting every aspect of my life. I was downright unbearable-- I'm still surprised that my now-wife stuck with me through some of my craziness. My dyslexia has always been relatively mild, though personally challenging. I'm very self-conscious of it, though its actual effect on my day-to-day life is minimal.
In order to be successful, I had to address and work towards correcting my issues.
If you haven't reached out for help with your social anxiety, either through medication or therapy or some combination or both, I highly suggest you start there. Though I have never gone the therapy route, Wellbutrin literally changed my life. To this day, I wish I hadn't waded through the years of self-imposed terror before reaching out to my doctor. Those were days needlessly wasted.
I also spent a great amount of time researching strategies to deal with my dyslexia, and worked to implement those changes. I've had more moderate success here, but it's had a huge, positive effect on my confidence.
What's more, I've worked in jobs that require me to do a lot of communication with both internal and external business partners. There's no substitute for practice, and I'd like to think that between addressing my own issues and having massive amounts of repetition, I've become relatively successful in the communications realm.
That said, if I could give you any advice, it would be this: do not wait to start your new career. If life has taught me anything, it's that there's never a perfect time to do anything. If you wait around for your personal emotional and mental well-being to exist in a perfect state to dive into the world of UX, you will never do so.
Start now. Work hard. You can absolutely do this.