i woudl recommend you stick with saying Shh - it is repetitive and more like white noise, whereas singing can be stimulating.
try to just get him calm (ie relaxed and not crying) but not asleep on your shoulder. if he falls asleep on your shoulder and then transfers to cot, when he goes through sleep cycles he will know he is not where he started and will need you to puyt him back to sleep on your shoulder, etc by putting him in the cot awake but drowsyyou are letting him learn to finish the job himself. he may not do it yet (and that is why in the beginning pat/shh is needed by some babies to get all the way to sleep) but eventually he will put himself to sleep and then he will learn to resettle himself and get through several sleep cycles, etc
if he is startling himself because his hands are free, then yes a swaddle may help.