Well, that's exactly how I felt. In fact, a couple of times I did leave baby with DH and run away . Always found my way home after an hour or two though - bit of space probably did us all good to be honest. My DH was also often the one to pick up DS1 to comfort him - DH was the calm, coping, practical one that seemed to be able to soothe DS1 while I was the sobbing milk machine crumpled on the bed wondering what the hell we'd done.
The lack of sleep is brutal and most likely at the root of the way you're feeling (plus the massive impact of all the hormonal shifts). I can't put a timescale on when it will get better for you as it's different for everyone, but it will. My DS1 is 19 months now and it's been better for a very, very long time. DS2 will be arriving 4 weeks today in fact!
Some things we did, in case they help:
Don't worry about the crying - I just accepted I would occasionally spend time with tears running down my cheeks and not necessarily know why. If you need to cry, let it happen, it's fine.
I expressed a bit of milk so DH could kip in the lounge with DS1 for a night and give him bottles while I got a decent sleep. Would this be an option for you at all? Expressing isn't for everyone, but it might help.
Similarly, not for everyone, but a dummy can help a baby settle I believe, if you don't use one already (haven't used one myself so not sure on advice round them).
I found that occupying my mind when he was crying in the night or feeding and I was exhausted, helped enormously as I wasn't thinking about how bad I felt and it calmed me. For me, I mentally chanted buddhist chants, but singing, poetry, anything like that would work the same.
Something I didn't do and wish I had:
Spoke to my midwife and / or HV. I should have done this and I didn't because I'm independent and I could cope
. Balls. I was a mess and should have spoken to more people about it. They might not have a magic bit of advice that sorts everything out, but being able to talk about it to someone that understands helps.
Chin up, you probably don't believe it, but you will survive and in a few months, you'll be sat in the sun, watching your DS roll around on a blanket while he babbles to himself and you feel relaxed and happy.