St Thomas is such a beach island - and lots of nice beaches. Maybe the best for having a choice in so many beaches.
As for shopping, what are you thinking? Trinkets? High-end jewelry or clothes?
(We did not grow up shopping on vacations and still don’t do much of it while on vacation. I am more likely to visit grocery stores to see what is different. I picture most kids getting squirmy in stores - while at home or on vacation. Actually, I have seen few kids in shops in ports. Or maybe they are quiet and well behaved).
The capital of St Thomas has an attractive harbor front which has been upgraded since Hurricane Irma (2017). A lot of shops are housed in centuries-old stone warehouses. Clothing, jewelry…
Tourists shops are also right at the port.
DCL usually docks at Havensight (West Indies Co - WICO). You walk past these when returning to the ship.
In some ways, St Thomas is less like what you would expect a Caribbean Island to be as far as villages. They have supermarkets and K-Mart plus chain pharmacies.
Coral World - you could do a little research and see if this would interest your family. It is next to Coki Beach. I saw a video where an adult couple visited this recently. It looked appropriate for the curious or young kids.
You could get on a taxi (you usually share) - and usually are open-air pick-up trucks converted, with bench seats and a top. Head to Magen’s Bay and then leave when you have spent enough time. Plenty of taxis are available for return. This is considered one of the best beaches in the world.
There is a look-out on top of the mountain above Charlotte Amalie. It has a little shop and restrooms.
There is also a gondola/ tram near the port. It will take you to a place to look over the harbor. It has a restaurant.