First off, UO, in my opinion, alcoholism isn't a character defect. Oh yes, I had/have many character defects and some of them contributed to my addiction, but alcoholism is a disease.
Why do think you need more time? I would be quite sure that your AV is giving you that line. In the face of treatment, your AV is going to be in the forefront doing whatever it needs to survive. Recognize it.
You are so blessed to have a family that is standing by you. And, believe me you won't be getting things handed to you. Stopping drinking and Recovering will likely be the hardest work you do in your life. Embrace your family's support and stop drinking.