We've done one Caribbean cruise but didn't spend anytime sunbathing, so can't really help with your direct question, tbh. BUT when the ship docks on shore days you are free to lie all day on the beach at the different islands! We did so in a couple of places including Grand Cayman which is perfectly set up for lazing on the beach!
On the ship, I think it is very busy around the pool on ship days, if I'm honest. But in the Caribbean there arent very many ship days - much more time is spent in port.
I'd recommend a curse to anyone, if I'm honest, it's the best family holiday we've ever had. But it suits us as we like to see and do different things rather than relax day after day.
We have never done a cruise but like the idea of seeing lots of different countries. We have 3 children 10,9 and 6 so we know we are probably looking at 2 rooms/2cabins. What are the pros and cons of cruising, can you still sunbath or do you have to fight for a sunbed?. Would love to hear your experiences - Thanks