I had no idea there was a Dutch Zeeland.
There are a lot of "Zees" in Dutch words. I have lived in upstate NY and Jersey (sorry, could not resist!), where there is a lot of Dutch influence.
There is the Tappan Zee Bridge, for example, that connects Rockland and Westchester Counties via the Hudson River. The Tappan part is Native American, the Zee from the Dutch. And in MI, where I grew up, there a town called Zeeland, near Holland, MI. The Dutch got around!
As far as calling the state I live in Jersey, that is pretty much what the residents call it. You will never hear, "the New Jersey Shore." In formal settings, definitely it is NJ. In casual situations, Jersey is the norm.