Caribbean w 3 and 5yo?

We are planning to take our kids for 4-5 days to somewhere sunny and warm in March, optimally 3-4hr flight from NY, and no ferries/yachts to resort. Not wedded to any island in particular, but want calm waters and a stay with walk out pool. Emphasis on good clean food - think grilled fish - hard to come by in a resort/family friendly setting where it's standard American kids menus (burgers and fries). I don't imagine leaving the resort too much given the kids ages but also want there to be enough for them to do. Suggestions?