Check out this blog, by a fellow vegan,and you will see how happy she was with Carnivals vegan options. Princess I would think, would give you similar options.
Vegan Crunk: Vegan Dinner on a Carnival Cruise
Both Carnival and Princess have good kids programs, with Carnival getting the edge. Might be fewer kids this year...Some kids are not out of school in early June, especially, in the northeast, with so many snow dates this year. Both ships have some nice beach options,and mix of islands.
Leaving closer to home is always a plus, so that may be the tipping point for you. Make sure your ta knows exactly what you need as a family to enjoy your first cruise.....we all here, know your kids will be constantly be smiling,and expect to want to book again. Good luck!