Buddy, I'm sorry, but you really are going to have to resign.
If this were just simple cheating on your wife with a mistress or even a steady gay lover, I might be inclined to think the matter should be left to you and your family to sort out. But gay indiscretions in a public men's restroom--no.
If you are stupid and selfish enough to solicit sex from complete strangers in a public men's room, with (seemingly) no regard for the medical safety or feelings of your family, then I don't think you deserve to hold public office, and I sure as heck don't trust you with power.
Then, of course, there is the issue that you've apparently been lying to your family all of these years about your preferences. Not cool--though I think you're of an age that 'coming out' is probably something you were taught never to do.
Granted, I don't have to hide my sexuality, and I've never been tempted to be unfaithful to my husband. I don't truly know or understand the strain of having to hide such an integral part of myself. But buddy, what you did was just wrong.
And that isn't even mentioning the Ewww factor. I mean, you don't know where they've been.