I'm in full transition and HRT.
I go to the doctors in skirt, bra/panties, whatever. (But always look good).
The only one "embarrassed" or shy about it was me, So I stopped and just went with my new normal.
I even had to go to the Urologist a couple weeks ago (to check out a strange lump in the nether region that disappeared on its own). I wore a skirt and panties as I normally do, and it made for easier "access" to my bottom area for examination. The doc also gave me a regular male prostate check (a freebie I didn't come for.) Never never said a word about how strange it was to check prostate on a woman.