Thanks for sharing your story. There's a lot of support here. Sounds to me like you've answered your own question - even limited drinking causes major problems for you. Unless you want to live with those problems ( and they will get worse most likely ) then you have to choose to stop drinking for yourself. There is support here, and many forums on secular recovery methods as well if you don't choose AA ( which by the way doesn't require a belief in god at all ).
Regarding your last paragraph, no one can give medical advice here so I'd suggest seeing a different doctor or a specialist to investigate further.
Take care and best of luck with whatever decision you make.