I'll disclaimer what I'm about to say by noting that I've only sailed Holland America, Regent, and Disney.

I think what you're looking for may best be served by Celebrity. Pretty much any short cruises to the Caribbean like that will tend to attract families, especially the cheaper lines like Carnival and Royal Caribbean. Holland America is excellent but you will likely be the youngest passengers aboard, but that didn't bother my wife and I at all when we started sailing with them in our late 20s.

Do consider Disney. Although clearly a family-friendly line, they go to great pains to separate and enforce adults-only areas on their ships. The ships themselves are gorgeous, and the staff provides that Disney level of service that is so difficult for other companies to match (except at a MUCH higher price point). Even on Disney's private island there is an entire beach and set of activities that are all 21+. The shows, food, shore excursions, etc. are all great. Now that I think about it, they may be the perfect cruise line for you if you're at all "Disney people"!