Well done OP. Glad you're seeing the light again. It's so grim when you're dog tired. Your mind sends you to dark places so easily then.
The DH seems like a decent bloke and those piling in to berate himself should check their own prejudices/projections. It doesn't do OP any favours to start attacking her partner now and blame him for absolutely everything.
Sleep deprivation is the absolute pits and impacts on both partners. When I look back at some of my not so finest moments when I argued with DP during those first punishing months, it was more often than not driven by absolute exhaustion. I wasn't making any assertive requests - I was just aggressively voicing how tired I was and how he wasn't doing enough hoping he'd step in. As it happened DP also slept badly and had to get up to go to work at 4:30 am so we were locked into competitive tiredness.
I ended up having a few sessions of therapy which did me a world of good. DD would come with me, sleeping or just sitting with me. My therapist just reminded me that nobody else is responsible for my needs but me and I just started communicating my needs and making specific requests. It made a world of difference.
"I need to have a lie in. Can you get up with her tomorrow please"
"I need to rest and have a bath - could you please look after her for couple of hours"
"When you come home I need to go for a walk alone"
"She won't settle, I've fed her so can you please take her out for a walk"
"I've booked pedicure for Wednesday and will be gone for a couple of hours"
"No, don't want to visit your parents on Sunday. This week has been really difficult and I want to stay in"
Etc. I know it sounds pretty basic but it's so different from
"I've had no sleep! I can't cope!" Or "You swan around doing fuck all and I do everything" or "I feel like I am the only one looking after her and you don't listen to me!!" or "Your parents don't get to dictate when we go there!"- I have said all of this and more at some point when on 3 hours sleep a night. 
And above all - mix feed if expressing doesn't work. A bottle of formula won't kill your baby and your supply is very clever. It'll adjust.
Good luck OP. All of this will be a distant memory before you know it. 