Detach from HIS behavior the rambling of a drunk on a computer screen and cell phone.
It’s documentation of his behavior that can be used against him so to speak.
Pick your battles, do you really want to battle with rambling words of an alcoholic? NO, so pick the things in real life, not threats on a computer to battle over.
I agree with wanting, the novelty of being a father will wear off soon enough because he will need to be responsible and that may be a big stretch for him.
It’s as if he knows he’s got you over a barrel with his threats and will continue until he is re-trained by YOUR NEW BEHAVOIR, other wise it will be the same old same old between the two of you. You have to change the dynamics of this relationship where he is not in control over you.