Brad: A lot of non-religious people have gotten sober, including me. There are lots of secular recovery programs out there. I attended a short 10-day inpatient detox/rehab and then created my own recovery plan that I dutifully followed every day. There was no religious component to it.
I don't mean to be critical here, but it seems like you are spending more time seeking out rationales to excuse your relapse, rather than a plan to quit. Cancer diagnosis - nobody can question drinking over that one. AA too religious - gotta keep drinking.
It is perfectly fine to be non-religious, but it is a massive flaw in thinking to allow that core belief to serve as an impediment to recovery.
I spent years creating excuses why it was okay to drink that one day or that one drink. It was only when I came to the conclusion that there is simply no good rationale for drinking and actively sought out and embraced a recovery program that I got sober.
Put down the drink now. Find a program. Check yourself in somewhere for a detox. Do something, right now, to get yourself started on a path to recovery. Don't waste any more of your precious life with addiction.