The trouble with getting grown/older kids to do their ownwashing, IME, is that just when you're about to pop a wash on yourself, you see that one of the idiots has got one t-shirt & a pair of jeans merrily churning around in there for the next hour.
I just have two baskets, one for dark clothes which fills up every couple of days, & one for light coloured clothes which takes all week to fill. If I notice a basket is full, I stick a wash on. In theory, dc do the same, but tbh none of them are great at noticing...so I get them to do other chores instead. If they don't put their dirty stuff in the basket, it just doesn't get washed & they can sort out their own load whenever they eventually run out of pants.
Having said that, it's not really about the washing, is it? If he's already stropping about that, he's going to be a PITA about everything else. You need a proper chat about expectations: getting a job, picking up after himself, & which bits of household stuff he's going to do.
Talk about food, too. I like to cook, so if I'm planning to I tell everyone on the family WhatsApp group that we'll be having <whatever> that evening. They're welcome to opt out, but if they're fending for themselves then they buy in anything special they want, stay out of my way when I'm cooking, & leave the kitchen as they found it. We don't have a rota for evening cooking because I genuinely enjoy it, but I do get brought breakfast in bed at the weekend!
Oh & think about money. Food, electricity, gas, water...do you need him to contribute? That definitely needs to be agreed before he's moved back in.