I've been a member of the same gym pre, during and post transition. The girls locker room is more chatty, but it is common for some girls to keep their knickers on while changing or go to the loo to change undies. So that shouldn't be an issue. As for wigs, I (fortunately) don't have that issue but I know our friend Catherine uses tape to keep herself safe and she goes swimming in her wig and on one memorable day on the Manly ferry it was blowing a gale, we were on deck and she had no accidental revelation!
From what some of the PTs have said I think I'm the only transwomen at my gym. I hire a female PT everynow and again for upgrading my routine and they have been as friendly as can be.
It is like everything we do Sis, look it in the eye, smile, and just be proud and be yourself.
I have nothing to apologise for being trans and neither do you.