I have to say that I'm impressed that you have got to the stage of submitting your work to magazines/publishers, and actually getting letters in response. Even if they are, so far, rejection letters. It's further than I've got, so far at least.
But, I'm not giving up, and I earnestly hope that you won't either.
Also, I hope that you'll forgive me for saying this and maybe stating the obvious, but I find myself wondering if you and I may share a tendency towards " All-or-nothing thinking "?
I feel really " knocked back " if something negative or disappointing happen to me sometimes, particularly if I'm feeling " low " at the time anyway.
So, may I ask you a hypothetical question, even if it might sound a little foolish? If the letter from F&SF had been an ACCEPTANCE letter rather than a REJECTION letter,would that truly have made such a great difference in the quality of your story or your worth as a human being? What I'm trying to say is, and perhaps not doing such a good job at it, sigh,

, is it possible that you are you placing too much weight on purely external validation
Your story doesn't become any less creative or valuable just because one magazine takes a pass on it today. You are not any less valuable a human being because F&SF passed on THIS story, THIS time.
Will your story suddenly be validated by publication? I would say that your story will be exactly the same when it's published as it is now, after this current rejection.
I would say that what is valuable in your story, and in you as a writer and person is intrinsic, it is already there. Now, today, this moment. Give yourself a chance to see the worth of your own work, and your own worth as a unique and therefore irreplaceable human being. And, give others a chance to see these things as well. Even , if it takes a certain amount of time..
OK, lecture over!
PS. What is " flash fiction " ? I don't recognise the term? Is it a particularly US term?