This sounds just like what we went through with my son, around the same age. He'd fall asleep fine in our laps or on our shoulders, but the minutes we put him down, he would scream his head off! He'd also wake up during the night, crying for attention and I'd only get maybe 3 hours of sleep at a time. We were desperate to find a solution that worked because we were so exhausted all of the time!
The answer that worked for us came from a website called www.sleepsense.net. We had to pay $49 for the information -- and I'm pretty sure it's all "licensed" and such, so I can't just link it here (don't want to get in trouble!) -- but I can tell you that it's basically a way to systematically "train" your child how to put himself/herself to sleep.
The first night we used it, it took him about 2 hours to finally fall asleep (you do things in steps, and have options as to whether or not you want to be in the room w/ your child), but then only about 45 minutes the next night. By the third night, he was out in 15 minutes, and slept through the whole night! We were sooooo relieved!
Overall, I would say it took him a week to 10 days to really adjust (still woke up at night a little bit, etc.) -- but then he was okay. Now, I can put him down for a nap or at bedtime when he's still awake (but I know he's tired) and he'll put himself to sleep. What a relief to have him self-soothe again!
I hope you will consider it. It was the only thing that made a difference to us and I'm glad we used the info she had.
Good luck to you . . . here's hoping you get a full night's sleep very, very soon!