I was in Cozumel just a few weeks ago, it is very safe, they cater to the tourist, without the tourist they will go broke in a hurry.
Belize I would never do anything on my own except the shopping area the tender takes you to. There are excursions there but I am a bit leary of them, you are not in America where we have safety standards, checks and rechecks. Several yrs ago a lady died needlessly cave tubing in Belize (on a carnival excursion), in Roatan, the zipline broke and killed the rider (On a ship excursion). This was several yrs ago and I am sure they are a bit better (maybe) just realize you are not in America when you plan your excursions