That can be so exhausting! My son was a lot like that. He usually fell asleep nursing, but so often he woke up when I put him down. I didn't think I was ever going to get anything done or any time to myself.
There were a couple of things that helped. During the day when he nursed to sleep, I would continue rocking him or holding him until he showed sings of being in a deep sleep (very quiet breathing, limp arms, that's all I remember). Sometimes I would place him in the very center of our bed, lie down next to him, nurse him to sleep and slowly roll away.
At night I usually just brought him in to bed with me, latched him on and went back to sleep myself.
It's tough, but it's also very normal to babies to continue waking to eat. Hang in there!!! Because it really does get better. Good luck :-)