Grand Cayman is safe. You can go anywhere on your own by just grabbing a taxi. You can do independent tours there to the Sting Rays and save yourself some money.
Just make sure you keep your watch on SHIP time. There will be a clock at the gangway as you leave that you can double check with.
However, Ochos is a whole different ball game.
I don't want to sound like an alarmist, however, we have walked the 5 miles from Tel Aviv to Jaffa on our own in Israel and felt safer than we did in Jamaica. We are known for grabbing a taxi in any foreign port (even when we don't speak the lanuage)...however, I would not do that in Jamaica. It is one of the very few ports that I would never do without a ship's tour (preferably) or at least an organized tour through a guide recommended on these boards.
Have a great cruise.