Well, you are in a mess. It sounds like your mother still has a major problem with drinking, and it's saddest of all for your little brother. Your granny needs special care. Once someone really goes downhill with Alzheimer's they can be a danger to themselves or others. Your mother obviously isn't up to the task of caring for her once you move, so yes, you need to get granny somewhere. You should contact the Medicare offices before DFACS. Your grandmother is on Medicare, so they should have several homes/assisted living centers or home health nurses (that's sometimes done -they basically have a "sitter" -often with nursing experience -to come in during the day when your grandmother would be alone). Hopefully you won't have to go through DFACS this way.
You have a choice when it comes to your brother. First of all -it sounds like he needs to get his license. Secondly, if you want and it's okay with your husband, you can offer for him to live with you. All of you ultimately need to sit down and confront your mom about the drinking. Let her know this is her wake up -that you're about to move out, granny is going to a home, and she can drink herself to death or seek treatment. Not by intention, but your being there is basically enabling her because she knows someone else will catch the slack, so she's seen it as a big party opportunity. During your intervention with her, you can remind her that she would have money if she didn't drink it away. Booze is NOT cheap! She may also be using drugs of some sort -and then the money really goes fast. Sorry you're having to deal with this, but confront Mama and don't let her condition get in the way of your family's happiness. Unfortunately some people really have to hit rock bottom and lose everything before they wake up.