Is South Jersey actually independent, or a protectorate of New York?
It's a protectorate of New York. The President of South Jersey is mostly appointed by the President of New York up until 2010, when it held its first independent Presidential election. Still though, it is under New York military protection and is considered to be in their sphere of influence.
Vermont is also a protectorate, but New York doesn't nearly have enough influence as it does in South Jersey. Vermont is largely independent of New York politics, but New York's armed forces defend Vermont from foreign attack. Vermont does have militias though for when New York isn't around to help them.