Hi welcome to BW forums
Great advice from Anna, as always.
I just wanted to add a couple of things:
He cries as soon as we put him down but when we pick him up he practically falls asleep in our arms.
If he is falling to sleep very quickly when you pick up you might be able to put firm hands around him in the cot and just pick him up a couple of centimetres off the mattress (this may be after a period of your regular PUPD so that he really is on the verge of sleep) and he might calm almost instantly in which case you can lower him back down and keep your hands on him, firmly so he feels 'held' but also feels his body weight on the mattress. If he cries again instantly your hands are already in place to do that little pick up, it's almost soothing in the cot.
I didn't use the full on PUPD method as mine was sleep trained younger with shush/pat but I did find myself in a position once (with a transition into a new cot) where I had no choice but to PUPD for a night, after a while my DS was falling to sleep the instant I picked him up, so I did as described above and literally pick him up a centimetre and put him right back down again.
It is also a step closer to soothing in the cot, firm hands or patting, rubbing/stroking in the cot are reassuring for LO. It's ok to do this if it helps him relax in the cot and it reduces how much you pick up. A firm hand on him in the cot can be weaned in the same was as shush/pat by reducing over time.
It feels like we are just teaching him that if he cries he'll get picked up.
This is absolutely what you want him to teach him - that when he needs you, you will always respond - through total confidence that you will always come when you are needed he will actually need you less. Imagine he is laying there thinking "oh no I am alone, Mummy is not here, maybe a big tiger will get me", he cries out, you arrive in a second and he thinks "oh, right, Mummy IS looking out for me, so even though I cannot see her, even though she is not holding me, she is still on guard and no tigers are coming, I'm fine, I'm safe." What you are teaching him is that he IS safe to sleep in his cot alone, that he CAN fall to sleep in his cot rather than in arms and that you WILL always always be there when he needs you.
My DH and I are just so frustrated because we were hoping this would work out.
Sleep training is a process, it really can be hard work for parents initially but the benefits are great, you won't believe it when you are able to put him down give him a kiss and say "night night, sleep well, call if you need me" and leave the room. It feels great as a parent to get that Y time but it also feels wonderful to have given your baby the gift of independent sleep. Although mine was ST younger I still have a couple of wonderful memories. Once after a tricky period of short naps where he'd needed some additional help and I felt like we were back to square one and had lost all ability to self sooth and independent sleep, I finally realised he didn't need the additional help but I had still been giving it (ie I was bothering him with extra soothing which he didn't want), I did our short wind down, put him in his cot, said "call if you need me" he gave me a most wonderful smile as he watched me leave the room. I popped back in about a minute later and he was sound asleep.
I recently supported on a thread where mum and dad were walking LO up and down for hours to get him to sleep. Mum and dad are now watching LO self settle on the monitor whilst they have (well deserved) tea and cake
Imagine! You'll get there.