I use women's restrooms. I started at within a month of starting HRT. I was afraid at first, so I would go to the furthest, most remote gendered restrooms, to avoid running into anyone. After about 3 months I started using the nearby women's rooms. Haven't entered a men's room in well over a year. So far no issues.

I changed the gender marker on my license within days of starting HRT, and got an amended birth certificate a few months after. I belong there, and if anyone objects I have the paperwork to prove it.

Can't speak for locker rooms, I don't like gyms or sports. I would feel uncomfortable in any locker room. So no, but restrooms have been no issue at all.