It would help if you could tell us the ages of your DC,s OP, but as you haven't I'm assuming you don't want to.
I've already suggested trying to have a night without the DCs...any chance of this? Not a date night, not even going out, just a bottle of wine with no kids and chance to talk without being interrupted.
All I can say, from experience of having four DCs, and from being an auntie to twins, is that it will gradually get easier.
I would really try and put a stop to you being solely responsible for the Dtwins. It seems that he has taken on the role of sorting out the older DCs, and that you tend to deal with the babies, but you run the risk of them never settling for your DH, which isn't really fair on either your 4DCs, your DH or you.
Maybe he could start off by giving the dtwins a bottle each while you read the older DCs a story, or cuddle the babies while you bath the older two..?
Maybe your DH and the oldest DC could get one of the babies ready for bed, while you and the other DC get the other baby ready?
If he doesn't get involved with the babies as much as he should, I imagine his resentment towards them will only get worse, if he feels that they don't respond well to him, or cry because they want you, when he's trying to care for them.
Like I said, it would be easier if I knew the ages, all of the above might be irrelevant, depending on the ages of the DCs.