The web site in the post below may help you in your beach searches. I found this in another forum:
Nothing against Chankanaab or Playa Sol, but as a full-time resident of the island, I can tell you that there is so much more.
There are lots of other beaches and beach clubs that do not charge an entrance fee.
Chankanaab is $10.00 to get in and the snorkeling is okay, but we like the Dzul Ha area much better.
Consider renting a car for the day and driving around the island. The east side is not to be missed and there are no tour packages that will take you there. Miles of white sand beaches and open sea. Only a few cabana-type restaurants. No electricity, no hotels, no beach clubs.
www.cozumelinsider.com offers an excellent beach chart for the entire island with info on the amenities offered at each location. You can then go to a separate page and read more about that beach as well as check out the photos.