Oh God, I remember these days.
First, take a deep breath and reassure yourself that the is NOT your fault. Ignore anyone who tells you you've created a clingy baby. Two of my 4 have done this. One was preemie, and the nurses in SCBU told me (on day 3, tearful with hormones) that the wonderful thing of a preemie baby was that the nurses could get them in a routine while they were still too tiny to know different, so you'd go home with a baby on a 3 hour feeding schedule that didn't need to be held/rocked to sleep and was used to sleeping flat on their backs..... well within a week the nurses all looked ragged as one of my boys would scream all night unless held so they could either have him in his incubator screaming, setting off all the alarms because his heart rate and blood pressure was rising, or they could wrap him up and pace with him all night...
Two of my 4 were like this. It's nothing that you've done or haven't done. The downside is, there's no quick fix. You have to ride it out. Survival mode time!
My suggestions:
Get a decent Velcro swaddle as others have suggested. Swaddle them, then feed and cuddle to sleep. Wait until last first sleep cycle (20 mins in your case from the sounds of it) before attempting put down.
When putting down, lie them next to you and have a snooze. Some babies need to Feel the warmth of your body. Also start using loud white noise, perhaps a heart beat or similar.
Use the swaddle for all naps too to build up the sleep association. So swaddle, feed, pop into pram for walk.
Do you use a dummy? Life saver for my worst sleeper. Some babies need a lot of help to connect sleep cycles. I used to put the dummy in after feeding (always in swaddle) and then cuddle or pop into pram. It took a couple of weeks but was AMAZING after that.
Take shifts. If DH has to work, you can't expect him to be up all night. But he can give you a 4 hour block at some point e.g. if your good at going to bed early he could do 8pm-midnight. Its amazing what a difference 4 hours makes. Also ask for help during the day. As baby sleeps in the pram, ask someone else to take them for a long walk so you can have a 2hr nap. Sleep is essential, housework isn't - it'll still be there in a few weeks! You can survive with 4hr at night and a nap in the day plus whatever else you can manage.
They will grow out of this. It's just about surviving in the meantime. Building up other sleep associations will help it pass more quickly.