Any place for people with fear of writing?
I do resonate with your experience because I was scared to write (English is not my mother tongue) and I am also an introvert. I have been reading much about writing and networking and here are some tried and working methods (by me) that I wish will also help you. In the case of real fear, I agree that talking to some professional will also be advisable.

Don't "write," instead, draft then edit

My biggest "A-ha--!" moment was learning that most writers don't really write a perfect sentence at the first go, instead, they draft and edit. And for me, my mind has eased up so much once I have learned this distinction and decided to split my writing tasks into drafting and editing. In drafting, I just freely interpret the data and plug in some discussion here and there, occasionally with some anecdotes, etc. I never edit them. Then, after a couple days, I return to edit the piece. On a good day, I got to keep about 40-60% of them; on a bad day, I may have to slash 80-90%, but also often have a good laugh at what I wrote. 

This finding really liberated me. I'd suggest you give this a try as well.

Write to think instead of think to write

Instead of writing to tell people what you think, I found it easier to write it as if I am having a discussion with myself. I wrote the research question, and then gave some answers, and then went back to question the answers more. I was often surprised that many times I read the scribble and figured out "Oh, this is what it is!"

I would advise against crafting out a perfect sentence and then put it on paper. This is futile because mind works a lot faster and non-linear than typing/writing process. By the time a "perfect" sentence is written, a lot of useful thoughts might have been suppressed or forgotten.

Dedicate times to write

Silvia in How to Write a Lot introduces a method that involves making a time and a space to write. I adopted the method this way: I block out time a few weeks ahead as writing/analysis time, then I guard that time. I cleared up the wall and the desk I face when I work on the computer so that I can only see the computer (and other books/articles I use) when I write. There is no picture, stationery, picture frame, etc. 120 degrees in front of my eyes. I also close down e-mail, silent my phone, and close the door.

Boice in Professors as Writers: A Self-Help Guide to Productive Writing even goes so far to suggest coupling important and crucial daily ritual (such as shower) with writing hour. No writing? No shower. I found this kind of stressful, but perhaps may work for people with a different personality.

Document time as well as words

Productivity aside, it's more fulfilling to document my writing effort in both time and output. High output is, of course, great. But I have also come to realize that reminding myself I have been dedicating a fixed time regularly to write, albeit slow and low, makes me feel more confident.

Free writing

Goodson in Becoming an Academic Writer explains some pretty nice workshops to foster a healthy writing habit. The starting one is to "free write." Free writing is not new, and some people use this to warm up or tune into the internal writer channel. It's really simple, just sit down, open an blank document, minimize the visible area so that you cannot see what you type, set a time limit, and pour out all sorts of thought that goes inside your mind. Anything, just type. 

Usually, after the time limit (I started with 5, now 10,) you'll have a pretty clutter-free mind and clearer idea on what that day's writing is gonna be.

Use proxy of writing

There are many, many ways one can put down an idea without initially writing it out. Drawing a mind map, popularized by Buzan, can help forming a bird view of an article. Paper and colored pen are good enough, online tools such as VUE are abundant.

Doodling, such as Sketchnote recommended by Rohde, can also be a fun way to "play" with words and idea.

For scientific writing, a paper by O'Coonor and Holmquist in 2009 suggests a pretty interesting method: basing your writing on a key output, such as a graph of the results, or a table of an analysis, and then build from there. For most experienced researchers this may be nearly common sense, but for those who think a paper should be written from Abstract to Conclusion linearly, this paper would help re-orienting them. My former supervisor, knowing my hesitation for writing, actually used this very method when advising me.

Use a recorder to document your thoughts or use speech recognition software to dictate your spoken words into texts may also help you break away from the deep-rooted fear towards writing.

Harness creativity

When writing output falls, I feel tired. After some trials and errors, I figured out that I feel tried not because I lacked energy, but because of too much creative thoughts didn't get to be expressed. So, I play music, pick a challenging recipe and make a potentially horrible dish (and eat it, butter + pepper + red wine always save the day), play LEGO, knit and crochet, come to answer questions... etc. I actually picked up a lot of habits through my doctoral study.

Introvert vs. extrovert

Being an introvert is actually not bad! In most modern culture extroverts get a lot more praises and attentions, but if you cast a critical look at it, this world needs both types to run properly. Two books I read last year were quite inspiring: Zack's Networking for People Who Hate Networking provides advices on how to focus your energy and properly carry out quality networking; Laney's The Introvert Advantage provides a more holistic look at how extroverts and introverts work, and how to cope and thrive as an introvert.

Closing remark

Best of luck and, really, enjoy the ride even it's scary. After the PhD thesis, there will be more collaboration and you will not feel as lonely. When I am down and want some healing reading, I often flip through Lamott's Bird by Bird and Zinsser's On Writing Well. They are also a good choice before reading those more serious and pragmatic how-to guides on writing.

Whenever I feel cognitively/academically inferior/inappropriate, I would think about a remark of Florence Foster Jenkins, who slaughtered Mozart's Queen of the Night aria (Try listen to Edda Moser's version first, and then Jenkin's version): "People may say I can't sing, but no one can ever say I didn't sing." Before worrying doing it well, do it first, and you will improve.

The fact is I have never met one person in my life who genuinely embraces writing as if they "love" to write. Most successful writers, when interviewed, often just say things like "What tips? I just write," or "I draft, and then I write, rewrite, and then I rewrite again." Betty's Drawing on the Right Side of the Brain has made a lot of people realize they can draw; I wish there will be a book called Writing on the Right Side of the Brain that will make people realize that we all can write, but we probably have to wait until the neuroscientists get over their fear of writing.