Ahoy, Tammy!
I am happy to hear that you have already acquired passports for your voyage in June! It looks like you are on the Disney Fantasy 7-Night Western Caribbean sailing. That is one of my favorite itineraries!
What to do with your passports on an excursion is a bit of a tricky subject. When we travel on our Disney Cruise Line vacations, we bring our passports whenever we are NOT on an excursion that we booked directly through Disney Cruise Line. But, on any DCL excursions, we leave our passports locked in the safe on the ship.
By taking the passports with you when not traveling directly with DCL, you have the insurance in case you have something unforeseen happen while in a port. I would not recommend cutting it too close, the ship will not wait for stragglers, and you definitely don't want to be stranded.
I would highly recommend booking your excursions directly through DCL, especially in Grand Cayman, which is usually a shorter day in port. Our family loved the
"Island Tour and Snorkeling with Stingrays" excursion.
I hope this helps with your cruise planning! Please let us here at the Disney Parks Moms Panel if you have further cruise questions.
Happy Cruising, Tracy V.