I agree with Nancy regarding Key West. Depending on where you dock in Cozumel, shopping can be a fun experience with lunch at Pancho's Backyard. Belize is going to be a bit more challenging, however. There is some shopping in the cruise terminal, but I guarantee that you will not like Belize City given your dislike for being approached to buy stuff.
I think I would recommend staying on the ship in Belize. The tender ride is long, the weather hot and once yuou get into the port, it doesn't seem to offer much for folks unless they want to go to the beach, cave tubing, fishing or otherwise participate in some activity.
Have a great cruise.