I think that you can be recommended a hundred different things and could end up spending a fortune and still having the same problem so I'd maybe try one thing at a time rather than feel panicked into all these different options. As he's BF, the cheapest and easiest to try is cosleeping, maybe for a nap today. It's very likely that the comfort and warmth of being next to and attached to you will send him to sleep as he has been on you. Then you can start to decide whether you can relax enough to get some rest, maybe even sleep whilst he is feeding and sleeping.
If that doesn't work for you, then you need to think about things that replicate the comfort and warmth he gets from being on you. That might include a dummy to replicate he sucking, things things that smell like you, etc. Babies aren't made to sleep on their own in a cot - that's our modern take on parenting. If you think about your baby's instincts, it's always to be with their mother. If you don't want to be with them all the time, then replicating those conditions may work.
If that doesn't suit, you can try other things that have been suggested - pram, bouncy chair, white noise, etc. But the movement ones are obviously only going to be suitable for naps, they're not going to help with nighttime. I agree with the PP that a sling for the daytime is a life saver, but it doesn't get you any extra rest. Definitely something to consider if you don't have one though as it gives your baby all those comforts of being next to you.
If nothing like that works, I'd consider GP, cranial osteopath etc but if your only 'symptom' is not sleeping separately from you, I would try other things first.
Remember that babies do need help to go to sleep. It's not like the movies where you just put a baby in a cot and leave it to go to sleep. Well it's not for 99% of people anyway!! Whatever tool you use to help your baby sleep also has to be taken away at some point, even gradually, so although you won't want to cosleep forever, you probably equally won't want to be rocking a 2 year old, or have a 4 year old who needs a dummy or white noise etc. My point is find what works for you and your baby, don't worry too much about the stupid 'rod for your own back' phrase! As with everything in parenting, you adjust the way you do things at the stage they need to be adjusted. If that makes sense!