If he was just your boyfriend and you were just dating, then I feel he would have no financial responsibility towards your children.
But if he has practically moved in, then he's not just dating you, he has joined your family.
you need to know very clearly how he feels about 'taking on' another man's child and you need to know why he has - to all intents and purposes - moved in if he does not want to be part of a family.
he may feel that your children should never get a penny from him or help or support etc, because they're your problem responsibility. In which case, he has no business being involved with a woman who has children. Or he may feel that he would only have any responsibility if you and he married and he was their Step-father.
If you live with someone who has children, then you have a responsibility to those children. It is a 'package', and if you can't or won't do that, then you shouldn't be in that relationship.
OP - you need to find out how he feels about your children. If he sees himself being a step-father to them, what he thinks his responsibility is, if anything, while he is living with you.
I really think you should step back and take a good long look at this relationship.
He may even resent your children. It happens. If that's the case, do you want to be with him?