As a traveller from the Great White North, I think I can help.
I always have to leave home in a jacket, sweater, hat, mitts, scarf, boots, etc. to get on a plane and land in Florida for a cruise. Secret is, layers.
I try to get by with a polar fleece jacket instead of a down coat, it's lighter to carry. Also, running shoes if the snow is not too deep when I leave! Then I layer everything else, sweater over a T shirt or tank top, pants, but carry shorts and flip flops in my carry on to change in the airport when I get there.
Another tip; I use a big "Paschimina" wrap as a scarf for leaving, a pillow for the flight, and a wrap for those chilly dining rooms or theatres onboard the ship!
As for sailing out of NY, you will need long pants, sweaters, and probably a jacket or scarf for the 1st day or 2. Then, you can shed the layers and enjoy the Caribbean sun in your sumer wardrobe.