I feel for you. Sleep deprivation is a form of torture, plus your hormones play havoc. I remember how tough it was when I added to my family each time. I don't think I even left the house basically until my babies were about 6 weeks old & if I did I was late, tired & looking like crap lol.
I had 3 bubs all within 39 months.. I think back now & think how did I manage? Having my 4th was a blur because I had 1 at school & another just starting kindy & some days I would think "will I ever sleep again?" I still get woken multiple times a night & my youngest is 20 months...But that's it isn't it? Us mums do manage. Some how we plod along & get through it. I know it doesn't feel like it right now, but it WILL get better. Like everyone else said, leave the housework & if someone has a problem with it tell them they are more then welcome to clean etc. maybe see if hubby can take your eldest of an afternoon to the park or something so you & bubs can have a nap?
I hope you start feeling better soon Hun. Big hugs x