You are ok. Your drinking might not be ok. That's not who you are.
There is no reason to be ashamed. You did nothing wrong. You did a very good thing for yourself and for your children by reaching out when you needed help.
Please reach out again. Find someone in the "real world" who can help you now. There are counselors and therapists who will help if you seek them. AA is a wonderful way to find people who care, but it is not therapy or a substitute for therapy.
It sounds to me like both AA and a therapist/counselor/other mental health professional would be good for you now. Of course, only you can decide. As you decide, please don't discount all the real experience so many people here are sharing with you because we care.
More prayers,
Joe
My sitch More importantly, Light A Million Candles