*hugs* It does get better. We have a baby too who won't sleep except on us, and we're still navigating the waters, but it gets easier bit by bit, and suddenly an hour of sleep ends up feeling like you've gotten 4, and then baby looks up at you and smiles after screaming bloody murder, and suddenly it's all worth it again.
FWIW I was at a pretty baby friendly hospital and I slept with her not on me, but I was flat on my back, with my legs spread a bit, and she slept in between my legs with her head pretty much on my crotch, so she was slightly reclined. She loved it and was one of the only positions she's sleep in. The nurse came in, and just stood in the doorway with her mouth gaping open. DH had to tell her there was no way we were moving a sleeping baby.
I can't imagine how sleep deprived you are mama. The day I went into labor, I got up at around 10am- which sounds like lots, but means I probably got about 5 hours of sleep that night. I didn't sleep all day, then got about 2 hours or nap time at dinner, then fell asleep at 1130- and woke up an hour later to my water breaking and contractions. I labored all night/day long, had her at 3:22pm, and then didn't get a chance to nap either, and she cried pretty much all night long. I finally napped about 10am the next morning... so I'd been up 48 hours with about 3-4 hours of sleep in total, plus laboring in there. That was bad enough, so I can't imagine how bad you feel. The sleep will come eventually.