City Island has several marinas
(in the bronx). Don't discount CT/New Rochelle, etc. Your trip into the Bronx from CT would me twice as fast as a trip from Sandy Hook, just because of the way the public transportation is laid out. Have a very good look at your planned route
from the boat to the Bronx in any location.
Sorry I don't have a specific marina to suggest, but they are all expensive and all about the same. Liberty Landing is one of the best and most convenient, though also a little more expensive. You will find none are inexpensive in the summer, unless you are looking to leave the boat on the hard
And yes. Huge waiting lists, even at some marinas.
Try asking around at Port Liberty - a condo development in Jersey City, South of Liberty State park. I know a guy that lives there. There are tons of unused slips and they are owned by condo residents who might be inclined to rent one out, assuming you are nice n yachty looking.