You definitely want to leave the port area as far as shopping .When you walk into port you will walk through a covered area with a number of shops. In that area vendors will stand outside and harass you trying to lure you in their shops. Very annoying!
We did the Mayan ruins, but the tour took us around the whole island. When we reached the backside of the island we drove by several beautiful beaches with very few people on them. Our tour guide said a lot of people will unknowingly stop and swim at these beaches not knowing the undertow is very strong and dangerous there. ( There are no warning signs there). As we continued around the island he pointed out beaches that were safer.(I would ask the locals where the safest beach would be.) We swam at a beach club, and the water was calm and clear. One tip: the ocean floor is covered with big rocks. I would bring water shoes to avoid stubbed toes. The backside of the island was almost uninhabited because that's where the hurricanes came on to the island. In fact it seemed most of the city of Cozumel is crammed around the port area. I honestly never found anywhere to shop, because everywhere in the city near the port were little shops where vendors were aggressively trying to sell you junk. I recommend a trip around the island with a guide or driver sharing the history. Also, if you can locate a safe beach I think you would enjoy that. We went in Feb. and it was in the 80's and sunny. The water felt great.