My heart goes out to you - your distress and sense of shame are palpable in your post.
Alcohol dependence is a very complex matter - alcohol does have addictive properties but more often than not it is not an addiction to alcohol that needs addressing but the underlying depression/avoidance of long-standing emotional problems.
AA is absolutely invaluable in helping people live clean and sober lives - I have a couple of friends who say they literally owe their lives to the fellowship they have received at regular meetings and by working throughout the 12 steps.
However I think it may be very important for you to have some one to one work in order to really explore what it might be that the alcohol is being used for, what it is anaesthetising. I am a counsellor and psychotherapist, and whilst I am not writing this post to suggest my own services, I am plugged into a large network of professionals, some of whom have extensive experience in matters of emotional distress and substance misuse. If you would like me to make any suggestions please PM me. Otherwise there are some many online directories - www.counselling-directory.org.uk
itsgoodtotalk.org.uk http://www.ukcp.org.uk/16/information/h ... -therapist
There is also the option of a residential treatment but this would depend on your ability to have help with your child, and of course finances - although many health care plans will cover a period of in-patient treatment
Try not to be too hard on yourself - think about showing yourself the same compassion and understanding as you would you offer to a close friend who is going through what you are.
Take care and please do contact me if I can offer any further assistance
I wish you the very best of luck for the future