You have received very sound advice from the rest of the hive and I only have my perspective to offer…
I am very sorry your Fiance has put you in this situation. However, I am relieved he told you and was upfront about his infidelity. It is important for you to fully understand that he has cheated. Internalize that. You repeat that he was drunk and he never drinks and you say he has no excuse, but it sounds as if being drunk is an asterisks to the betrayal. Yes, drinking can impair a person’s judgement, but to what degree?
Having said that, one of my closest friend’s Fiance cheated on her 7 months before their wedding and was in a very similar situation. (He was drunk and hardly ever drinks. He told her immediately after it happened. Everyone thought he was a real stand-up guy and loved his personality and sense of humor.)
It was very hard for her to get through that time in her life. She confided in only a few people, but before you know it – the rumor-mill started and everyone knew within a few weeks. After that… everything changed. Her friends and family were protective of her and basically shunned the Fiance, which caused her to run to his defense. Then it became them against “us”. Family and friends started to talk behind her back about how weak and naive she was. As a result, she looked to her Fiance for support. Since they both needed each other for social survival, they never really hashed out their problems, concerns, etc. The actual incident and feelings were just “glossed” over and continue to plague their relationship. They have since been married and although she is happy 75% of the time, she has turned into what you fear you will become. That wife that doesn’t trust her husband and therefore won’t let him do anything.
From your post, it seems you are fully aware of how dire this situation actually is because once you tell one person (ONE) chances are that person can’t keep a secret like this to him/herself (very unfortunate).
My advice is this (and has been recommended by other bees): get tested, go to a therapist (alone) and then go as a couple. Go ASAP. Just start the process. Make an appointment today. I know therapists can be expensive and time-consuming, but in the long-run it may mean the difference between being happy and unhappy. Also (you probably know this already, but it needs to be said), he must be held accountable and responsible for his actions. However you see fit, hold him accountable.
As for the wedding, would it be possible to postpone? Publicly, you could merely say “health issues” are the reason for the postponement (and yes that is true – the emotional health of your relationship is an issue). Privately, postponement doesn’t mean you won’t get married; that just means the wedding is pending. You both have to repair the damage he has caused before you can commit to marriage. And the damage may be more expensive than you think as most people cheat because something is wrong with the relationship. That doesn’t necessarily mean there is something wrong with you or between you two. There may be something wrong on his end as he may suffer from serious insecurities, fear of commitment, inferiority complex, childhood issues, etc… Alcohol is often a determinant for infidelity, not a catalyst. Please make sure you get to the root of the problem.
Well, I wish you all and only the best. Please take of yourself physically, emotional, and mentally.