If your partner has a major drinking problem/alcoholic/can't be trusted with the baby, then be realistic - this isn't working. It would be one thing if he faced up to it - but he's not being honest. This isn't safe for your baby. What are you getting out of it? Sounds like you're stressed and not getting any help.
For now, I wouldn't leave baby in his care. Take baby with you when you're running errands. Single moms do. Alternatively, leave baby with friend or family member and just say they want baby-visit time.
Don't argue/nag/confront him at this point if he's not facing up to this. He doesn't sound like he will change now.
You can. I'd consider what your options are. Stop focusing on the relationship 'working' and focus on you and the baby. He's not putting your family first. Maybe he's incapable of it at the moment. Maybe he's a true alcoholic.
I think there are moms on here who can advise you on how to proceed on that. My advice is to put you and babe first. Don't leave babe with him. That's just not smart/safe at this point in game. If you do need to while you're in house, etc. just strap baby into a bouncy chair - whatever you need to do so that baby doesn't fall of couch, or get smothered.
ETA - I agree with TF - these questions aren't making sense to me lately, and don't sound like written by a mom. The issue clearly isn't how do I leave my baby with this man, but my marriage is in trouble - I'm married to an alcoholic. If this were real, that's the far bigger issue. You would never leave baby in his care.