OP, you could be me posting, in many ways. DH and I don't really row - it's more quiet and empty even than that.
I feel compelled to say it isn't your fault because you are exhausted and therefore grouchy. How's that your fault? There is no blame in that. The promenade clearly lies in how you as a couple deal with the sleeplessness and spread the pain. So in a way it's your DH's fault! He should be picking up the slack, and isn't.
But we are the same as you. I've slept barely 2-3 hours a night for 10 weeks, and when DH takes the baby off my hands at 5am for me to sleep, he has to make up the tine later in the day, so that I have both kids while he grabs three hours' sleep. Which of course I don't get to do. And that makes me pretty mad .
Like you it feels like I do everything and when DH does something it still is down to me to finish the job or clear up. Today DH gave DS2 a bath and I had to run up with the towel and then later empty the bath, put the dirty sleep suit in the washing basket, put he dirty nappy in the bin and wipe up the spilled water. And same if he cooks. It's dropped food and dirty dishes everywhere and way more to clean up than if I had cooked myself. It is very hard because it feels like nothing ever happens without my involvement. I've put it to DH as "sometimes I just want someone else in the driving seat while I'm the passenger" but it never happens. So of course I'm always either nagging or just putting up with it and letting the resentment build up. I feel like I am powering everyone along and nobody is putting petrol in my tank, IYSWIM.
And so I end up thinking it would be easier if DH wasn't here. It would be clear that I am doing everything, without the frustration of wanting him to do things that he isn't doing and without the extra work he creates.
I went to my GP on Friday and admitted I am going under. She offered the usual medication which I declined because I am breastfeeding. But she has also put in for urgent CBT for me, which I hope will help with the negative thinking hat my situation is creating. Maybe you could ask for the same?
No substitute, of course, for improving your relationship, but I, like you, don't have the energy to figure out what we should do and he sure as hell won't! Our communication goes further down the pan with each passing week as we no longer know how to enjoy time together and we are in separate bedrooms as I am co-sleeping with difficult DS2.
It is notoriously difficult to keep hold of who you are when you have kids. Your old "self" disappears in a vortex of chores and demands on you. It is a difficult period for any couple. Like another poster, I would urge you to hold off any drastic decisions while you are lost in this mire. And yes, I am taking my own advice and hanging in there until things are a little easier, and then I will reassess.
In the meantime, to stop things sliding further, I don't know what we should do?!