I feel for you AND the kids.
I thought i'd put my tuppence worth in lol.
I had something of this sort of .. i know you feel like you wanna strangulate DH and DSS.. unfortunately its illegal..
I know you don't WANT to appear to hate the child, just because he is a kid, doesn't mean he is easy to love, but you are the adult so have to grit your teeth and make an effort.
Well done for trying to involve him in the cooking thing..shame it backfired.
Maybe its just not his thing, or he is feeling so lonely, stressed, unwanted and angry that no matter what you do, you could not possibly hope to do the 'right thing'
DH is being a bit of a nob..but i think he too has assumed you hate his son and wants to make up for that.
What a tangled bloody mess.
IMO..you need to get to know this little boy, and learn to like him and enjoy his company, for it seems that your own son is picking up on your unhappiness and it is also making him angry at what he sees as the reason for his mums' upset, and the upheaval in his life.
Remember as obnoxious as he is, this is a little boy.
Tell your DH this weekend HE is taking out your DS and DD, and you take DSS out.
Ask him to choose something he would like to do.
Try and wheedle in the opportunity to have a chat with him and say something along the lines of, i really do like you but i think somewhere along the line we have got a bit lost. I don't think you are happy with this either, so we need to work at getting on. I would like you to be able to come to me if you are unhappy, ect. I WILL get on with you and we WILL be happy together, but it needs us both to work on it. I am not trying to replace your mum...
You get the drift, whatever you feel you need to say, and make sure it comes from the heart.
Just remember, although this is a child, they CAN be manipulative, as well as anyone else, i am under no illusions on that score, and would not slate you for disliking him, or he you, but putting it on open show is out of order. Yeah, he has been uprooted from his mum. He probably worries about her or feels unwanted. So he has moved in with you and you feel like you have two needy angry upset boys and no support or understanding from DH.
For this you have my sympathy.
And yes, he probably thinks if you disappear from the face of the earth he will live happily ever after with his dad.
He doesn't realise that won't happen..and he doesn't know the impact it will have on him and your whole family, he is just thinking as a child would.
What about your DD? Do you think all of this is worth depriving her of a proper family, which, with a bit of work and give and take from all parties, you and your DS included, because no one of you does not carry a little of the blame here for inflaming or adding to the situation, the only innocent party is your DD probably.
Spend time with him, get to know him, lay your cards on the table, and hopefully he will realise you are not the wicked stepmother..despite (sorry about this) your apparent recent efforts to prove him right..
And you will realise he is just a hurt frightened kid, not someone with the agenda set at pissing you/DS/ anyone else off, and of splitting you up.
I hope i haven't been too harsh. I am speaking as a stepsister, a step child on both mum and dads side, and having had a SM who was adept at showing her absolute hatred of us, to the point my dad lily livered out and stopped coming to see us for over ten years.. I believe i have enough experience.. but not trying to make you feel awful, i am sure you feel that anyway hun.
Wishing you the best of luck whatever you decide to do.