I'm 55. They've been awkward for me my whole life, and for the same reason as you. Thankfully, I have a very good Dr. who know that my anxiety is sky high during the exam so she keeps me talking about other things to distract me. Things that I do for fun, my dogs, work, etc. This helps to calm me down and reduce the anxiety.
Try to keep in mind that this is the profession that they chose. Bodies, any human body, does not "gross them out". It's just a body to them. They have seen them all!
If things like that grossed them out, they really could not do their job, could they?
Try your best to not think about it (way easier said than done, I know). It's all OK.