Obviously, the best outcome would be for you both to sit down and talk about this as mutually respectful adults. He can explain to you how he feels when you turn down his offer and you can explain to him how you feel when he sulks.
Your dh should understand that an offer of sex is just that - an offer - you are under no obligation to accept. He should therefore realise that sometimes you might decline and prepare himself for that so that he does not feel so personally rejected. He also has the right to decline offers of sex from you.
However, you say that you cannot talk about this as it always ends in an argument, so that leaves you with two options a) allowing him to have sex on you or b) putting up with him sulking.
Actually, allowing him to have sex on you is not really an option is it, so that leaves you with the sulking.
So, my answer to this problem is to stop the sulking and there is a very simple way to do this.
Remove yourself from it.
No-one sulks without an audience.
Every time he sulks you should leave the house. Go for a walk, go for a coffee or some breakfast/lunch/dinner, go window shopping, catch up on some errands, visit a friend, whatever.
Leave him in the house with the children and tell him to ring if he needs you to watch the children so that he can go to work, or whatever, then you return and he leaves.
Either way you are not with him and he will not sulk.
When he asks why you keep leaving the house, tell him it is because he sulks if you are there. If he says he is not sulking, he is 'hurt and rejected', tell him that you are willing to talk to him about it when he is ready but you are not willing to argue about it.
So until he stops sulking and stops arguing and gets to a place where he can talk about it like an adult, you will leave him to it.
Try it.