Requesting a change to conference lodging before accommodations have been verified?
The best way is just to be direct, though you may need to reveal a bit about the reasons. Explain the situation much as you have here. I think most conferences would try to accommodate you, but it might not work out in every situation, due to numbers and money. 

But do this immediately, so that it doesn't require reassignment of rooms, etc. 

Alternatively, if you are willing to pay for your own accommodations and feel that is worth it for preserving your privacy, you can just decline the offer and make your own reservation. It might be best to work with the conference, however, as they may have access to special rates for attendees. 

Universities in the US are not, in general, allowed to discriminate against people for medical conditions, so I wouldn't be overly concerned about revealing that you have a medical issue, leaving out details as you see fit.