I did the same loads and felt equally as awful I the early days. I spent hours googling and got all sorts of answers, the majority of which were just "don't do it, baby will suffocate/fall" and I just found them unhelpful as I didn't feel that was an option as she wouldn't be put down. People didn't talk about safe cosleeping.
One important thing I found out later is it is so common, and most of my mummy friends had the same but we were too scared to talk about it. Ultimately if that's the only way they will sleep, and people say "sleep when they sleep" but you can't cos they are with you, WHEN CAN YOU SLEEP?!
Can daddy have a turn holding so you can sleep? Some people take turns holding and sleeping.
Look up safe cosleeping on the WHO website, and on the lullaby trust. Obviously baby going down in their cot is preferable but cosleeping is natural and practiced I'm most other cultures. I have found a way to have my arm all around her so she still feels snuggled but we are lying down and I can see her airway is clear etc.
Read up on the forth trimester if you haven't already. Baby is getting used to the big wide world. It's a scary big place!
If nothing else, reckon it's still safer to prop up in bed as you are in a way that if baby rolls they won't fall (you would wake up as soon as they move anyway) and if you fall asleep you won't slump onto them. This is v important in avoiding the suffocation risk and preferable to sitting, especially on a sofa or chair where the risk of sids is highest. But really try to follow the safe guidelines properly - it would be devastating if something happened...
Eat snacks! Bf burns calories anyway and it will help get you through. And drink water. Play crappy games or scroll through Facebook/Instagram/mums net (which I discovered in those early hours) on your phone.
It will get better but baby may sill go through phases of waking a lot as they have various leaps/sleep regressions. Mine was sleeping through some nights at 3m but now usually wakes twice at 8m. Some nights it's more and in those cases she comes in with us!
Good luck and try not to beat yourself up.