tequila tasting and beach

My BF and I will be in Cozumel in December. We want to do a tequila tasting and decided on Hacienda Antigua. I would also like to go to the beach and maybe get in a tiny bit of shopping. The two beaches that look most interesting to me are Playa Palancar and Punta Morena. I want a beach that is not too crowded but also has good, inexpensive food and drinks. Which beach would be better to go to after the tequila tasting, and what would be the best way to get around to these places? We absolutely WILL NOT rent a car. Punta Morena doesn't look like it's far from Hacienda Antigua, but would we be able to get a taxi back to the other side of the island? How much might it be if we have the driver take us to Hacienda Antigua, wait for us, and then over to Playa Palancar? And how long would that take? I don't want to spend a fortune on a taxi, but I'm hesitant to book a driver for the day because we usually like to be flexible about our plans in port.

Hacienda Antigua also has a shop that is right downtown just off the plaza. We tasted all of the tequilas there and there was no charge.