OP, few mums are likely to be in your situation which is quite highly unusual. I can't imagine how stressful it must be and how hopeless you must feel to look after your two daughters and this pre-adolescent boy you'd never met before, all this is a small house.
You definitely help from professionals. For a start, you really need to pick your battles. Him not doing the dishes properly is not that unusual for his age. He is doing them, and that's already something. Don't get cross with him for this, focus on the positives which is his behaviour and more amazingly, his work at school.
However exhausted and frustrated you feel, try to put yourself in his shoes. Imagine being an 11yo, with hormones all over the place, rejected by your mum, taken away from the only family life you know, to be sent to a new family you've never met, who might have totally different habits and customs to what you are used to, whilst going in a school that you used to go to but with friendships that have changed since. He must miss his mum, his siblings, his friends.
What he's been through is quite horrendous and he is probably doing his best to just get through another day. Don't excuse everything, but try as much as you can to sympathise with how hard it must be for him. I know it must be very hard.