Those first few weeks are so tough. If she's falling asleep at the breast, she mightn't be all that hungry. Can DH walk the floor with her for a while so you can get some sleep? DD was similar at that age and I'd be about ready to drop at 4-5am. I'd get DH up and he'd rock, pat, walk DD around while I got some sleep. Even just 30-60 minutes can make a big difference to how you feel. Sometimes it meant DD would be crying that whole time, but it was much safer than me trying to care for/feed her when I was about ready to drop.

I know it doesn't feel like it at 3am but it doesn't last.

Also, have you tried taking her to bed with you? She might be after some extra comfort.