You do not owe him anything. It doesn't matter if he drank once a week, every hour, or not at all. Instead of listening to him, keep following your intuition about what is best for you. Verbalize what it is your gut is telling you. Maybe even right it down. It is hard for me to change my thinking patterns if I don't have words to put with my new feelings or intuitions. It was even harder when my xah kept coming at me with his words - which were marinating in emotional manipulation. I'm not a big fan of journaling but I did spend time writing some things down when I was at the point you are at now. It helped me to be able to have words in my head for what my gut was trying to tell me.
As far as him and the baby, what will be best for her? Throught out her life? This is not a time to get side tracked by feeling sorry for him. Play the tape all the way through. Alcoholism is progressive. It is not your job to cushion his sadness or take care of his emotional being. That is HIS job. Your job is to take care of *your* emotional being - and now that of your baby.
We each find our own way but I could not be friends with my xah. I hope that we will eventually have an easier relationship then we have now but I keep our communication strictly limited to the kids. I make small talk when I drop the kids off at his house and he is fine, but he can't leave my driveway fast enough. Whatever. I think he just wants me to get all flustered, worry about him, worry about what I have done wrong to cause him distress, and chase after him to do whatever it takes to make it all alright for him - because that dynamic worked for years.
I know in my soul that my only hope, and the best chance for my kids to have a healthy childhood, is to keep my life seperate from his. I will protect that by keeping our communciation to the necessities regarding the kids. ETA: I say that with great conviction now. It keeps me strong in the no-contact department especailly when my co-dependent self wants to be 'nice'. I have that conviction now. I wanted to let you know that when I was at the point that you are at now, I did not have that conviction. It is so hard to think straight and trust ourselves when we are in the middle of it but I will say - the clarity came fast once I had a little space and distance. Keep making the next right decision, trust the process, and give yourself a real gift. The gift of some time and space. Even if you go no contact for a month - that will help. Nothing is lost after a month. You don't have to be afraid of mistakes. We can always change our minds - the real trick is figuring out what we know.