I like
Frida and Edvarda, as well as
Irene and maybe
Irena. But it’s fine to change your name to something different from your birth name if it's what you want! There's nothing fake about wanting to feel comfortable and like yourself.
I think
Edwina could also be nice, though it doesn't actually have the 'eed' sound (but I think you could get away with using it anyway!). Things disconnected from
Edie that sound similar to me are
Edwige,
Iva,
Ivy, and
Iseult, though depending on where you live
Edwige and
Iseult may be difficult for people to pronounce.