for the diapers, make sure his penis is pointing down and that the little frilly-isa pieces of diaper on the thighs are pulled out. those were the two biggest things that caused diaper leaks for us.

the fussing at nighttime could be just normal baby fussing. it could be gas, and if that's the case you could try doing some tummy massage and bicycling his legs. other people suggest gripe water for gas, but i've never used it so i don't know how it works. you could also try bouncing him on an exercise ball, if you have one. neither one of my kids enjoyed being rocked or held while i was stationary, but they would settle if i bounced on a ball. if he's overstimulated, white noise can help. there are good apps for it and youtube has several 10+hour white noise videos. have you tried comfort nursing him at night? it was normal for both my babies to basically nurse on and off for several hours at nighttime, usually between like 6-10pm, before they'd finally settle down for a little while.

no matter what your baby might need, your SO still needs to help you! if he doesn't feel like he can get the baby settled, just tell him that you need a break, get a little time to yourself, and then everyone will be more equipped to deal with the fussiness. maybe remind him that sometimes all a baby needs is a change of scenery and having someone else hold him can do some good. most importantly, it's never to early for him to learn some ways to soothe his baby by himself!