If you really want to soak up some of Cozumel's real culture, take a taxi from the port (about eight bucks) and tell them to take you to Sabores for Enchiladas Mole with the purple Jimaca (pronounced Ha-My-Ka) pitcher of drink. It is cheap (6 bucks pp), incredibly delicious, unlike anything you have ever tasted and you will see that many of the locals chow down here. It is also within just a block or two of all the shopping and haggling you want. Not very far from the port. Extremely safe!!
There is a website I highly recommend created by ex-pats (mostly Americans) who live in Cozumel. It details many of the great places they recommend you go and so far, I have found them to be extremely accurate! Here's the link: