Agree with aiden, forget the party. Say you feel ill if you need another reason (sleep deprivation makes you feel ill so its not far from the truth).
I felt exactly the same as you - I had an 18 month old and a newborn. I was hideously sleep deprived and had no idea how I would cope. I cried alot and sometimes wondered why I'd had two so close together. I'm still up alot in the night but I don't feel awful anymore, I'm actually enjoying it. However, I did have the support of my parents close by.
So - can anyone help you so you can nap for a bit? Or if your body clock is screwed to the point where you can't nap, at least rest?
Sleep deprivation is so horrible - it affects how you feel. But it won't harm you, it's something you just have to endure in those early weeks and it really does get better. Don't worry about how you will cope, because you just will. And once you've managed on your own even for half a day, your confidence will go up and that will make everything seem easier.
Stopping breastfeeding isn't giving up, you have to make the right choice for you and your baby. But maybe give yourself another week and see how you feel then? It's such a lovely thing to be able to do when your baby is a bit older and you don't feel so knackered.
Also the cluster feeding stage passes quite quickly. And I was the same about feeding in public, I wasn't comfortable with it in the early days.
Sorry, keep thinking of things to add! I just know exactly how you feel. Things to help cope with hardly any sleep - drink lots and lots of water, eat as healthily as you can. And try bachs night rescue drops to help you relax at night, even if you don't get much sleep. And remember it always gets better!