We love the Western (won't even consider an Eastern, although going to the Baths on Tortola was a fun day).
Cozumel: we love Chankanaab Beach Cozumel. We've been there twice. It's easy to book on your own. Grab a cab and go - plenty of cabs to get you back to the ship as well. First time we did the Royal Dolphin Swim and loved it. Second time we did a swim with a sea lion. Chankanaab is a great place to spend the day. After your encounter (and included with the entrance fee) is a sea lion show, a pool, great beach with chairs/huts/hammocks, and snorkeling. A casual lunch or a buffet lunch is available. You don't have to do any of the dolphin/sea lion encounters. You can just go to the beach and still use all the amenities mentioned above and see the sea lion show. It's really a great place to spend a full day. We've been twice and still haven't explored it all.
https://www.dolphindiscovery.com/cozumel/?utm_source=google&utm_medium=mybusiness
We've also gone to Paradise Beach twice. LOVE their pool. Great beach and food. Not very crowded. Again we just took a taxi there and back.
One of favorite excursions ever was the Sea turtle release. It was an amazing day. I highly recommend it. (May only be offered certain times of year.)
In Falmouth we did the Heritage Walking tour and then had amazing jerk chicken in town. (I don't think this walking tour is offered anymore.) We've also gone tubing down the Rio Bueno twice which was a blast but I don't think
DCL offers Rio Bueno.
We've been to Grand Cayman a few times. We've done an island tour, gone to the Turtle Farm, and gone to Seven Mile Beaach.
Western itinerary is our favorite - best excursions - but it's just as easy to walk around the port or grab a taxi to the beach. Lots of shopping in Cozumel and inside the gated port area in Falmouth. In Falmouth they have a trolley tour that lets you see a bit of the island. It's at the end of (inside) the gated area. You don't have to book - you can just get a ticket that day.