I have read a lot of parents saying when they're baby sleeps through the night, they don't sleep much through the day. From what info I've read, you need to see their sleep as a 24hr thing. You may find your baby is getting enough sleep so cat naps is what his body needs to get through the day
Definitely read the sleep cues and keep a routine so change nappy, feed and then bed. If you can't rock him to sleep anymore, sit next to him while his in the cot and put rock, pat him or just lay your hand on his body. That can help to know you haven't left him and still getting comfort.
Also as pps have suggested, a carrier of some type will help a lot to get things done.

As for your DF, communicate and make a list of your daily routine. Men see things physically, so showing him it in writing will click. Also, offer suggestions on how you try to settle bub when he's unsettled.

You may just need to let your son be outta your arms while you do something to. Put him where he can see you and do what you can. Go to him when he gets upset but if just whinging, tell him he's ok, mum is here and even singing to him so he sees your happy can help.
Have a shower or bath together.
Lay him on the floor near the toilet so you can go.
It is ok to do the essentials, ie toilet and shower, bub will be ok even if crying as it won't be long that you're doing those things.