I'm sorry you're going through that. That is very frustrating. I think many, many of us deal with similar problems. Most males just aren't pre-wired for the job like women are. They have to LEARN it. Try and talk to him about it when you are both in a good mood and just ask for a little more help. You are exhausted and you just need a good night sleep. It would help you out SO much if he would volunteer to take a night.
My DH finally realized how much help I needed when one day, I laid our DS on the floor to play and DS started to fuss. I completely lost it. Like... I have never been so out of control before in my life. I burst into tears, ran upstairs, screamed "GET THE F**** OUT!!!!!!!!!!!!!!!" to the dogs when I got into our bedroom and SLAMMED the door as hard as I possibly could. I sobbed for a good 45 minutes.
It SUCKED that it got to that point- but it did. Now my DH is very careful to ensure that he is pulling his weight as much as possible. That incident scared him and it scared me. I have never felt so out of control like that in my life and I never want to get to that point again. It was from lack of sleep and feeling like I was "doing it all".