Yay for honeymoons! Congrats on the upcoming nuptials!
Grand Cayman we took a cab (they're mini vans) to 7 Mile Beach, entered the beach at a public access, found a quiet spot and relaxed for a few hrs. We then walked along the beach toward town for a bit, got back on the main road, hitched a cab back. Then spent the rest of our time window shopping...tons to see!
Cozumel we rented a car for the entire day and it was WELL WORTH the money! We got a Tracker for about $75. There are cheaper options (mopeds etc) and I bet you'd save $ if you made a reservation before you left on the cruise. We used Hertz. We had a great day travelling around on our own and it's impossible to get lost...there's only 1 major road around the whole island. Most of the beach clubs are free to use, so you could stop there for a bit...Paradise Beach is the newest and looks beautiful. We stopped at Nachi Cocum which was also nice. You can travel along to the east side of the island where things are more underdeveloped...beautiful, crashing waves, lots of private spots to be romantic. Drive on and stop at "Coconuts" for a great, affordable lunch ($30 including 2 huge margaritas!) with the best view of the coastline from its location on a bluff. You could then go tour the ruins at San Gervasio...I think it was about $6/person to get in. End your day shopping and strolling thru San Miguel...the center of Cozumel.
Hope that helps! Let me know if you have any ?'s.