You've made a bit of a rod for your own back!
Charge him rent. If you don't really need the money, put it into a savings account so it can go towards a house deposit in the future.
I think it's probably easier to do the whole family's washing in one go, but get him to do one wash a week. That includes sorting, loading, emptying as soon as its finished (he'll just leave it in there to fester for days otherwise), hanging out, folding, putting his own clothes away and everyone else's on their bed. He needs to consider the time it's going to take, and choose a time he can complete the task to the point of hanging it out.
He can certainly load the dishwasher or empty it. Or wash up/ dry up.
Stop changing his sheets fgs! If he wants to sleep in a stinky, sweaty bed, let him.
Teach him to cook. He can cook with you/DH for a while, let him choose some recipes he likes the look of, then once he's more confident, he can be head chef with you as sous chef until he's able to fully prepare a decent meal without help.
With £1600 spare money a month, he can afford to move out. I wouldn't allow to do nothing and if he won't help, he can leave.
Your DH is completely unreasonable. Why on earth wouldn't he want his son to be independent? I hope he's not a sexist pig who thinks a woman should just take care of all these things for him and DS for life.
These days DS will struggle to find a GF or wife (or BF/Husband) who'll happily run around after him like you do without complaint.
And once babies come along, men need to put more effort into the household chores. Without being asked, preferably.
Please don't allow him to become a useless, thoughtless, selfish partner. No future GF/BF will thank you and he'll struggle in relationships.