Okay:
1) Your Husband is still acting as a solo man. He is NOT acting nor thinking like a "parent who has a baby", yet.
So it is TIME he get it and steps up. How?
2) YOU TELL HIM WHAT HE NEEDS TO DO.
3) He has a BABY now... and YOU need to tell him what to do, because he is not doing anything... and it just going about his OWN things.
4) At what point.. is he going to REALIZE... that he has a baby and he is now a DAD... and ALL OF THIS REQUIRES HIS PARTICIPATION.
Period.
So since he does not realize any of this, you need to tell him.
Hopefully in a calm manner... because once a woman starts "nagging" a Man shuts down and acts like a toddler.
5) DO NOT... treat him like a "hotel" guest. He is not.
TELL HIM, what you need him to do. WRITE it down and post it up on a poster... and on the wall.
The problem is: you are doing everything. But... you are not telling him... what he is responsible for.
The thing is: you'd think this is all COMMON SENSE for an adult "man"... but for some guys, you need to spell it out for them.
NO ONE, is "exempt" from household duties/child care and rearing/chores/errands/cooking etc., once they have a child and are a family.
AND YOU NEED to, on your calendar, just TELL HIM, when you are going out for whatever reason even if by yourself, to do whatever you want and to get a break. AT those times, HE NEEDS to care for the baby and the house.
And since he does not seem to know how to do these things: you need to TEACH him how. He needs to LEARN... HOW TO BE A PARENT AND A household member.
Show him, how to bathe baby, how to change diapers, how to watch baby so that he/she is safe, CHILDPROOF the home, HAVE him read books on it, etc.
HAVE him do these things while you are at home, so that you will know that he is doing it and learning it, NOW. So that, when you go out.... you KNOW he will and can do it.
Its about time, he realizes, he has a BABY. It is HIS, too.
Not getting any sleep, once you have a baby... is how it is.
Your Husband, HAS TO REALIZE, all of these things.
Sleep and having your own time, is hard to come by.
Even when a child gets older.
For the 1st year or two, not much sleep is attained by the parents.
Child development and phases, is how it is.
So, your Husband, has to CHANGE is outlook.
It just is.