Disclaimer: I live in a suburb of a liberal haven. People aren't exactly trans-friendly here, but they've got enough exposure to the city that they're ashamed of intolerance.
I'm a poor grad student, so I go to thrift shops for all my clothes. The first time I bought clothes from the women's section... well, the first dozen times... I was pretty nervous. I'd circle the store, wait until the section I wanted was clear, take a circuitous route to the fitting room, wait until nobody could see me there... gah.
I've gotten over it. Whether I'm presenting as male or female, I'll just head on to the section I want to go. Women in the aisle with me? Whatever. Men walking by while I'm trying to put some sweet knee-high boots on (no, they don't fit

)? Whatever. Some people will look. Some people will care. Those are the people whose opinions matter the least to me. Most don't care. Most won't look.
It can still be a little nerve-wracking, for sure. But I do it anyway, and it gets a little easier every time. My favorite coping mechanism is to just keep my eyes on the racks, and don't look at people.
When it comes to bras... my chest is about the same size, and I just got an A-cup pushup bra. I went to La Vie En Rose. Some of the employees looked at me weird (I was presenting as female). One came over to help me. I focused on her, and she was super nice and helpful, and I didn't bother to look at the other employees. This is a good technique in general: hang out 'til somebody comes over. The employee who approaches you will be comfortable with you.