Welcome to SR. You are singin' our song. We know exactly what you are going through.
I have no doubt that your AH is a wonderful person. But he is a sick person, first and foremost. First of all, try to remember that relapse is a symptom of the disease. It can and does happen.
Next, when you argue with an A, they have this amazing ability to throw up smoke screens, make up excuses (unhappy with work, etc.), and turn things around so that everything looks like our fault. They will say just about anything to achieve this. Don't you fall for it!!! This is their way of getting the focus off of them. And for a while, we even believe them ("Is it ME?" "Am I making a big deal out of this?" We ask ourselves). I think we do that primarily because we want the problem to disappear. We want to be able to dismiss and/or downplay what's going on. But it gets worse, and worse and worse, until it's SO in-your-face that you can no longer ignore it.
I have witnessed what you said about how he appeared to get drunker and drunker before your very eyes, even though he never left your presence. I can't explain it, but I have seen it. We went to dinner one night with my daughter and son-in-law. He was perfectly fine when he got into the car. However, the farther we drove, the more he started slurring his words and not making any sense. When we arrived at our destination, he literally fell out of the car. It was as if he had been drinking before we left, and the alcohol had a delayed reaction, or something.
My AH's choice was also vodka. It was so easy for him to hide this in a glass of water, for example. They get extremely creative in finding ways to drink without your knowing it. But you DO know, as you said. I could tell the second I laid eyes on mine. One word out of his mouth, and I knew.
My AH would disappear for hours. To this day, I don't know where he was.
About AA... it is NOT focused on God. It is focused on sobriety. There are members of AA on this forum who can explain this to you. In my opinion, rehab and AA are your husband's best chance. My suggestion to you is to learn as much as you can about the disease of alcoholism so that you are armed with the knowledge you will need to face this. You should try attending some Alanon meetings for yourself.
Since your husband has expressed a desire to get sober, he might be open to entering a treatment facility. I would, also, learn more about AA and approach him again with this idea.
Lastly, here's one thing you can cross off your list of things-to-do: You can stop checking his breath and looking for clues that he has been drinking and is lying to you. He has and he is. It's as simple as that. Confronting him will not accomplish anything, except create more turmoil in your home. Believe me, I know this is a lot easier said than done. You can't talk him out of it. You can't reason with him. And you need to save your energy for more important things, like taking care of your baby. You can't have any more "alone" time, because your husband can't be trusted to stay sober while the baby is in his care.
Please read the other posts, and you keep posting.