Unfortunately, it's really very common and typical for newborns to want to nurse constantly. Breastfeeding isn't just food for them-- it's also comfort in a big world, and connection to you.
The best way I now how to handle this is to work on learning side-lying nursing, so you can sleep while baby nurses. That's how all of mine slept as young infants. So if you haven't begun working on this, that might be a good idea. I used to just latch them on, slide down to a side-lying position, and go to sleep myself, and they would nurse until they fell asleep soundly. Usually I was asleep first.
It can help to put a small rolled-up blanket behind baby's back, to help baby stay in a side-lying position-- you can tape tightly it to keep it from unrolling.
Are you swaddling tightly? The only way my kids would allow me to put them down asleep at this age was if they were tightly swaddled. My DS STILL wouldn't let me put him down, but it's worth a try.
Also, in a few weeks if you are still struggling with this, you might try the very limited use of a pacifier. The paci can help you ease baby off the breast and stay asleep if put down. You want to be cautious with it-- if baby stops nursing well and often, you'll want to put it away for awhile, and definitely don't use it to delay a feed for a hungry baby or to avoid meeting baby's needs. But you know that, I'm sure.
This stage passes very quickly, when you look back on it later, but when you're in the middle of it, it can seem like it's going to last forever. It won't!
Hang in there.
If you can nurse side-lying, one thing to try is to scootch over into the cosleeper, beside her, and nurse her until she's sound asleep-- then often you can gently ease your nipple out of her mouth, and leave her lying where she is, and go back into the bed yourself. I had some good luck with that strategy, with DS-- but not until he was a bit older, I think.
