ok... been to Coz several times, and from experience i would stick to the western side of the island for beaches. most of the beaches on the eastern side are rocky, and some of them can be quite sharp(not a place to take a fall). the farther north you go on the eastern side the beaches become a little more sandy and less rocky. Chankanabb is a nice place to spend a few hours on their beach, but they have much more to do than simply sit under a palapa drinking a cold drink... they have the dolphin encounter, a small zoo? if you want to call it that...gift shops, etc. there is a cover charge of about 26 USD i think, just for entry into the park, and more for other activities you can choose from. as for shade, you shouldn't have any trouble finding shade at any of the private beaches, and a couple at some of the public ones. usually in the form of a thatch roofed palapas or larger covered bars and pool areas. if you have the time you may want to check out Coz on tripadvisor....