I am not an alcoholic but I do have depression and have dealt with similar dismissal from some family members when I told them. Honestly, I think the only way to get through is to continue calmly educating them. You have to act in control, even if you feel angry and out of control, so that they will listen to you. If you explain that it is an addiction and a disease, and that you would appreciate their understanding and support, that's really all you can do. If they refuse to believe you, that's something you can't control.
I felt judged, especially when I decided not to start medication after all (because I wanted to try therapy first), and this decision seemed to signify to my aunt that I was therefore not sick. If I wasn't taking meds, I had no disease. This only made me feel worse about my depression, feel like I was being weak by not being able to overcome it alone, and guilty about even thinking of taking meds.
So I feel your pain! I hope your family can listen to you. It may take them some time to come to terms with the idea.