But, they can always give it to the state AG if they want.

The only thing they have him for, presently, is the federal campaign finance violation. Unless NY has a similar enough law that they can tailor to the facts they've already established.

They'll have to give Cohen something to cooperate also, nothing happens in jail for free. and since he's serving a federal sentence, not sure how much pull they'll have with him.

violate both state and federal law. If Barr says to no to federal prosecutions, the state can take over. So yeah, they actually can just hand it over to them, assuming it's even necessary. From what I understand about the SDNY, they pretty much have autonomy in how they conduct their investigations. The only thing they need from the DOJ is permission to indict based on their findings. Permisison denied? Hello, state AG!

There's the NY AG and the Manhattan DA that might as well (bring their own indictments).
There is cooperation you know.