Debbie, with that budget you can cruise for a lot more than a week.
You said you wanted to cruise the Caribbean and leave from NY in April. Does that mean you want to SAIL from NY or that you want to FLY from NY?
You can in fact sail to parts of the Caribbean from NY, but not in April. A little later in the season (May) you can sail out of NY on an 8 day to the Caribbean on a Carnival ship
(probably Carnival Legend) and you can sail out of NJ (Bayonne) on RCCL (on a 9 day
it's either Voyager or Adventure of the Seas). You can sail out of NY year round on NCL
but they only go to Florida and the Bahamas. You can also sail from NY from late April to October to Bermuda on any number of cruise lines from Celebrity to RCCL to Carnival.
Your choices out of Florida and other southern ports are too endless to begin to list.
I suggest a good Travel Agent specializing or with experience in cruise bookings.
Once again your budget is extremely generous to say the least and you should be able to travel for MUCH less.
Welcome to CruiseMates and Happy Cruising.