Have you gone to Al-Anon? You are trying to control her drinking which can't be done. At Al-Anon you will get support and see others who have the same problem, which will put your situation in perspective. Unfortunately many people in relationships keep hoping the alcoholic will change for the better, but the alcoholism doesn't go away so change is unlikely for many.
The most important thing in an alcoholic's life is the bottle, so she isn't the only person who puts her family second. She prefers to continue drinking, so unless that changes and she accepts her alcoholic status and tries to get help likely nothing will change. You are likely powerless to try to turn her into a different person.
Sleeponit, did you know that alcoholics are born with a genetic marker for becoming an alcoholic. Once they have one drink they can't stop drinking. That doesn't mean some people do it because they like it and have an addiction. But most alcholics like your wife (probably) cannot stop drinking unless they get help and have to not take one drink ever again. So they have to get the mentality that their first love (alcohol) is their enemy not their love. How you change that takes a lot of AA meetings and at this point her liver toxicity his high and she has to have help. What I see with my brother is similar. I haven't seen him in 4 years. He does call/write once in a while but he lies about everything. He won't talk about the drinking or he will hang up. He has had 2 DUI's that he admits to. My uncle had this my grandpa, and my mother, and my brother since he was maybe 17. He hid it very well. I didn't really know until much later. I caught him stealing my stuff and I thought it was for money for his girl friends. He cheats on everyone, gambles (another family addiction). But, I doubt she would get tested and it isn't FDA approved yet). My husband is working on this test at a biotech. But it is a fact. So, don't beat yourself up. Try to take care of your kids. Support them all the way and it is a hard job for a single parent. That is what you are. Sorry to say that. In the mean time have you gone to Alanon or I'm sure how to spell it those help groups to understand the mentality of an alcoholic? They will be one all their life even if they quit. They also have a "persona" of who they want you to think they are. It is so hard on you. They can come across as super cool or a educated person with no problems until the crap hits the fan. The main thing is she needs professional help. She has to want to get it. How you do that? I haven't figured that out. You have to not help her get her alcohol or give her money in any way. Maybe tell her she cannot see the children until she quits. Harsh love. Sorry. It is not good for them to see her like this. I lived with a drunk uncle. I loved him but we hid from him when he came home drunk. Not because we were scared but didn't want him to bother us. it was not a good memory of that side of him we felt shame, then my brother followed what he did. A model for the kids to see. That it was okay to drink and gamble like he did. So my brother did exactly what he did instead of going the other way like I did. I don't happen to have the gene so I was lucky. Meaning I have had some alcohol and don't really care about it much.
Hope you can get professional help here.