I was raised in an HMO, and never got the message (from parents or docs or society) that I should keep anything from them. I have a condition that has led me to lots of time in doctor's offices, and a fair share in the ER. I also worked in medicine, granted, as a clerical, but I learned a lot.
I've never hidden anything. When I was living with a woman for the first time, and we were exclusive, my doc kept asking me what I did for birth control. I kept saying I don't need it. Finally, I explained why I didn't need it.
When I received care at the university medical center I worked at, I was a little more shy about telling stuff; then again, I didn't have much to tell then. I have noticed that I tell everything and they seem surprised, but appreciative. (I had two med students, a resident, and a faculty doc at my last female annual. Oh yes, every single one them wanted to examine me. Happy to share. Pass that around.) They expressed their appreciation that I would tell them all that. Hay, I want the best care, they can't give me the best care if they don't know stuff.
All that said, if all you've got is a cold, there isn't a scrip for that. I get really annoyed at the hundred signs in my doc's office that say 'don't take antibiotics if it's a virus' and I tell him my troubles, or that I need a note for work, and he gives me antibiotics. No culture. When I was a kid, they cultured to see if it was bacterial before they gave antibiotics. (and I walked five miles, uphill, both ways, in the snow)(sorry)
If you've got a university medical center, they should be mostly LGBT friendly. If you've got a local LGBT anything, they should have some referrals. It might not be likely to get one for all, but go you if you do!
(I've been home sick for two days, you've got all my sympathies. And if you're interested in natural cures, I personally think GSE (grapefruit seed extract) is a miracle.)