Strange that none of the regulars has responded to your question. You will certainly need a jacket for the first two days. Even once you hit Florida you can hit coolish weather--we docked in Miami when it was 33 degrees F. It seems you have to go about another 1000 miles south before you are guaranteed hot weather. It certainly complicates the packing when you may be going from winter snow to tropical heat. You should see us running through the snow at the airport in our shoes and light jacket when we fly. Even more hilarious on our return (remember the movie "Cool Runnings" about the Jamaican bobsled team landing in Calgary). We have learned to leave winter jackets and boots with whoever is coming to pick us up, or in our vehicle at the airport.....chilly!! Once aboard ship, you will likely want to wear more than shorts, tank and flip-flops for the first couple days.
I boarded wearing a tee short & long sleeve blouse under a cotton cardigan. The sweater came off once on board. The cardigan was later worn over my shoulders at dinner when they turned the AC up too high.
I brought a few sweaters like that. Make sure that they can be worn over your more summery clothes.
You do want to bring your winter coat -- you will need it for embarking & disembarking. You may also want to wear it if you go on deck for sail away. Sturdy shoes are a must for that.
fleece layers are your best bet, you can wear a fleece jacket under your winter coat for getting on and off the ship in NY. You will need pants for your first couple of days at sea out on deck. IF your fleece has a hood thats even better.
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.