So with a little more digging, I found this place called Folklore Salon. On the site itself, it says "For Dames, Gents, and Folks in Between" and all of the stylists appear to be queer. There is even one that I believe is somewhere on the trans spectrum because they said "I'm a Southern California girl/boi".
So I'm gonna give this place a shot this week, unless I get any different recs here!
I'll let you guys know how it goes, post pics, etc. If I'm satisfied with the cut, then I'll recommend it to others that may be looking for the same thing in LA.