Welcome Closetdrinker.
I'm not a medical professional, but I'd say a six pack every night is more than binge drinking. Only you can decide if you're an alcoholic, and since you are here, looking for answers you obviously want to do something about it.
Can you look for another doctor, maybe one geared toward addictions?
As for the exercise, it sounds like running was helping you to not drink. Can you replace it with something you are able to do such as yoga or walking?
If you feel like meetings would benefit you, don't hesitate to go. You are worried about being recognized which is understandable. Maybe try a meeting the next town over where you may not be recognized? Honestly, maybe you wouldn't be recognized at all. There are only a handful of elected officials in my area that I would recognize if they were standing in front of me--and that's mostly because we just had an election! Then again I get most of my news from the radio rather than TV... just a thought.
You say you haven't had any legal issues or marriage issues. You left out the YET. If you keep down this path, you will eventually. Better to stop now before the YET can happen.
Good luck!