Hi Emma
That sounds really tough but also completely normal and, in fact, very common. I know that doesn't help you get more rest, but hopefully it will put your mind at ease as a starting point. Having your first baby can be such an enormous shock. My second is now 6 months and behaved very similarly to your little girl in the first 3 months (and even now). My first was the same!
You don't talk much about daytime naps but I recommend using a baby carrier. A nice soft fabric one will work best at this age. Your baby is in the 'fourth trimester' and, as tiring as it is, she wants closeness, warmth and motion. Can you get her to take a dummy? The ultimate combo would be to feed and burp her, pop her in the carrier with a dummy and possibly white noise and walk around in a natural way. It may take a wee while but she's guaranteed to drop off eventually.
As for nighttime, it sounds as though you are doing everything right. Honestly! In the first 3 months babies have very immature digestive systems and can find it hard to settle after a feed, as frustrating as that is. I think it sounds good to feed, burp, change, feed: I would continue doing that for the time being.
You'll probably find she sleeps best for the first part of the night and then struggles after her first wake up. So maybe you and your partner could swap over sometimes so that he gets the tougher morning hours too?
The reality is that you are doing nothing wrong and will probably have to ride it out. Things will get better at some point, although I can't tell you when. In the meantime, get as much help as you can. Let the house get messy. Eat takeaways. Soon you will be getting 4 or 5 hours' sleep and (although it's still rubbish) you will be able to get through the day without feeling so desperate.
I really hope that helps a bit. I know how you are feeling because I've been there. When you are living it, it is utterly horrendous and you visit some dark places in your mind. But one day it will just be a distant memory.