I think Pixie would be a cute nickname for a more serious full name, like
Patricia or
Priscilla, maybe. I like it as a nod to the illustrator of the Rider-Waite tarot,
Pamela "Pixie"
Colman Smith. I don't like it as a full legal name, it conjures up that whole "manic pixie dream girl" trope.
Fairy is still used a slur in some places, I think, although it's old fashioned.
Fay is ok, but
Fae has a more "fairylike" energy to it.
Fay is cloaked in a bit too much vintage dust to seem spritely.
My favourite "fairy" name is probably
Peri.