Can an employer deny employment to me because they think I have a drinking problem?

I was offered a decent paying job a month ago, only to have the employer yank the job offer after my background check came back(I have 2 drunk driving convictions, one 3 years old, one 5 years old and one public intox from 2007 - all are misdemeanors). They did tell me that the job offer was contingent based upon a satisfactory background check, but now I find out that alcoholism is a covered disability under ADA. My local civil liberties office says I have a case, but I would like to hear other opinions from people knowledgeable about this area of law. - The job that I interviewed for did not involve any driving, so my driving record can not be held against me and I do not have any other crimes on my record besides the ones listed above.