Well I'm not a woman but I am a man who wears womens clothing exclusively so I simathize with your situation. I buy a lot online but also buy retail. I've used the womens dressing room at Macy's once with no problem but there was no one around.
White House/Black Market is a women's clothing store and they have dressing rooms at the back of the store along both sides. They are all separate with closing doors and are located just off the lounging area at the back with a wall of mirrors. Very classy and comfortable. The women there are so sweet and helpful! I went in there and shopped for myself just like I was at a menswear store. They made recommendations on styles, fits and sizing. They even brought me additional things to try on that they thought I might like. The minute I had walked in the store, they knew I was shopping for me. Highly recommend them and places like them.
Regarding the response below, I don't buy the safety arguments. Women's dressing rooms are like mens. Neither are communal dressing areas. They each have there own separate cubby's with closing doors. No one walks out of their cubby in their underwear or naked, they walk out clothed. Germs are eveywhere, they exist in both dressingrooms. As long as you are not rubbing your naked body all over the dressing area you're probably going to be okay.