I'd be awful careful about discussing bras and showing your bra straps at work with your female coworker. If your relationship with her ever goes south, you could end up with her making false harassment charges (and being the male, you would be assumed to be guilty and treated as such.) She could try and threaten to out you too. Just sounds like a bad idea.
Skip the bra at the doctor's office. Unless it is important for them to know you wear a bra (and dress) or are fulltime, there is no benefit for them to know, and it might make them uncomfortable.
