Take a look at RCI Explorer of the Seas cruises from NY to the Caribbean, then book cruise only, book flights to suit before and or after the cruise and check out Tripadvisor reviews on NY hotels or better still get a recommendation from a friend. Shop around, we've managed a 12 night cruise, virgin flights and two nights room only for a couple in a top hotel in January bor £2200.
Why not book 'cruise only' to New York & then spend as much time as you like before flying home. Alternatively what we do is fly to New York, book a hotel for as many nights as we wish, & then book a cruise from New York. A holiday does not have to come from a travel brochure. Decide what YOU want to do & then present your plan to a Travel Agent.