Hi both, biggest sympathies, it is really hard, unfortunately what you are describing is totally normal. I have a 5 month old, who like AM, is a crier, and "this too shall pass" has become my mantra!!
Miss I'm not sure what to suggest as you've probably tried everything already. What about motion, does that help at all? E.g. car rides, being pushed in the pram (this could just be back and forth in your hallway/living room if in middle of the night), bouncy chair, rocking cradle, being held and rocked in your arms on or on a rocking chair, being held and bounced on an exercise ball etc? Movement helps my DD fall asleep. I find a lot of the time it is trial and error. Eventually you learn what kinds of things help them to fall asleep, but even then, that doesn't mean things will be a surefire way to get them to sleep, I often have to try 2 or 3 different techniques in a row to eventually get her asleep. It is tiring, you have my sympathies. I find that certain motion things mean she will fall asleep somewhere that isn't on me so that means I don't need to move her to put her down, because that often wakes her, as you say. Pram is great for that because then they're asleep in there you can just leave them in it! If you have a detachable carry cot and you push them in your hallway at 3am til they go to sleep, you could detach the carry cot and take it into your bedroom and get back to bed. We've also had some success in the evenings putting her in a Grobag. She was usually fine falling asleep on us then we'd move her to her crib, tuck her blankets in and she'd stay asleep for bed, but about a month ago she started waking about 50% of the time when we were putting her in her crib or tucking the blankets in, so now we use Grobags and put it on her before she is asleep and since doing that she hasn't woken once when we've put her down for the night in her crib. Might be worth a try if you haven't already? Other things too like white noise etc, I mean, anything is worth a try really! Also in terms of practicalities in the day time, have you tried wearing them in a sling or baby carrier, that way you have your arms free and they can stay asleep on you if necessary.
AM If you want to talk to some people who know what you're going through with the constant crying, find out 'Parents of Criers' thread in the 'Parenting' section. There's a few of us on there who support each other because we all have babies who cry all the time.