I don't find it so much embarrassing, but rather I do feel more vulnerable in those situations than if I were totally naked.
One example was a screening physical I had before I participated in a medical study. The assistant went over a questionnaire with me in the exam room, then she asked me to undress down to my underwear before she had me step on the scale to be weighed. Standing there just in my briefs I did feel vulnerable, for sure. Then, she said I could take a seat again, and the doctor would be in to see me. When the doctor came in, also female, she sat and asked me some more questions, then took blood pressure, checked breathing and heart, eyes, ears, etc. And, just as the assistant came back in, the doctor asked me to move to the exam table where she checked reflexes, motor skills,(I guess,) and a few other things I don't know the terms for, really. Throughout all this my sense of vulnerability in front of the two of them was certainly heightened. The doctor then asked me to remove my underwear, so I stood up and slipped my briefs down to the floor. When I stood upright, now completely naked, the vulnerability was much less, and they were now the ones who seemed to be walking on eggshells around me. Even while the doctor was doing the hernia and prostate checks, with the assistant looking on, I didn't feel quite as vulnerable as before with my briefs on.
The nudity changed the dynamic, as it usually seems to. I most often feel that I'm the one everyone else is deferring to whenever I'm naked in a situation with clothed people, but it's different when I'm wearing underwear in front of these same people. It's like my thing about wearing nothing but socks, I feel more exposed than when I'm totally naked. Definitely, that little bit of clothing makes me feel more awkward. I think it's that I'm not quite to that point of full nudity, so I'm still thinking as a partially, (or, barely clothed,) person, which is different than just being nude.