You're not going to look like a flake for quitting your job. If you've only been doing the part time one for a few months, you can leave it off your resume completely and just say you quit your full time position to stay home with the kids, or it was a temporary family leave, and you now are looking for a new position (in the event that you do still want a job somewhere eventually in a year or two).
edited to add in a few relevant links:
http://www.askamanager.org/2013/02/job-searching-after-being-a-stay-at-home-mom.htmlhttps://money.usnews.com/money/blogs/outside-voices-careers/articles/2016-01-18/8-tips-for-parents-returning-to-the-workforceSometimes jobs aren't a good fit, and you do not have to stay there just because you feel some sense of obligation towards someone. You are not a failure. Neither is your friend. It just isn't a good fit. And much like a jacket that you thought looked lovely in the store but got home and discovered it is is too tight in the sleeves, you take it back and part ways cleanly - the job/jacket was clearly not meant to be yours, and it's not anyone's fault or responsibility to keep it/wear it.
As far as the friend that referred you to this job...if they are your friend, then it shouldn't be that terrible to tell them that you're so grateful for their getting you on at the place, but you are unfortunately having issues with the isolation and giving presentations cause you anxiety, and you're going to be turning in your notice. When you give your notice, you could offer to stay on through project X or offer to stay a week or two extra (over the traditional 2 weeks).
It's serendipitous that this article just went up by Ask A Manager's Alison Green over on Slate:
https://slate.com/human-interest/2018/05/is-it-normal-to-get-incredibly-nervous-about-telling-your-boss-youre-quitting.htmlThe odd thing about all this anxiety is that quitting a job is not usually the bomb-throwing act people fear it will be. To state the obvious: In the vast majority of cases, quitting your job will be met with mild to moderate disappointment and a meeting or two to transition your projects. Then everyone will move forward and adjust to the idea that you’ll be gone (often more quickly than you might prefer).
I do hope you'll give yourself some time without work commitments to detox and just learn to be YOU for a while. If you were really burnt out from your full time position enough to quit, it likely felt quite disconcerting to suddenly be without employment. No matter how in touch/savvy we assume we are, suddenly not having a job to define who we are is confusing and downright anxiety provoking once you get through the first few days/weeks. You need time to just sit, read, piddle, stare at the walls, go to the library, the movies, shopping, volunteer, and play. Most of all, you need to find things that make you happy and content. If that means eventually, after many months or even years, you go back to (paid) work, then do it because you find the work itself fulfilling, not because of the money or any other obligation.
And there is nothing wrong with getting a part time job doing something you think you'd enjoy that isn't in your field of expertise if you don't want to be a full time stay at home mom (which is also completely fine too). Work in a bookstore? Volunteer at the local resale shop? Animal shelter? Local elementary school aide? YMCA? Get a PT job at your favorite store (so you get a discount)? I volunteer at my local library, teach a free yoga class weekly and do animal rehab for a wildlife rescue. I am currently debating about volunteering for a really cool charity resale shop that apparently offers a killer 25% off (!!) for volunteers. None of this has to go on a resume - it's for fun and for social interaction and for a sense of community and adult conversations. :D
Good luck!