I did a 21 night trip around Sicily and still couldn't fit in all I wanted to see. Regardless of 13 or 16 nights you will have to prioritize and limit what you can see.
If you fly into Catania and base there for three nights you can see the city itself, Mt. Etna area and a daytrip to Taormina. Taormina's very popular but its Greek theater and beautiful scenery are truly beautiful. You could easily get to Taormina as a daytrip on your own by bus or train and maybe do a paid daytrip tour of Mt. Etna area. A car is not needed in Catania itself but could be helpful if you want to visit the wineries and Aci towns between Catania and Etna.
Then head to Siracusa and stay in Ortygia. Again easy bus and train transport to that area so no car needed. You need at least three nights there. You can use Siracusa/ Ortygia as your base, lots to see and do and do a small group one day tour to see your Baroque towns of Noto, Módica and/or Ragusa. From Siracusa either hire a driver or rent a car to take you to Agrigento with a stop to see Villa Romana del Casale and maybe Caltagirone with its famous steps and pottery. Stay in Agrigento two nights. You won't get their till late afternoon anyway so that gives you one full day to see the ruins at Agrigento. From Agrigento take train or bus to Palermo. I would stay in palermo at least four nights, maybe 5. You need at least two days to dedicate to Palermo itself. and a few hours in Monreale. Then you can join a small group tour or hire a driver to visit Segesta and beautiful mountaintop medieval Erice. You can also do an easy daytrip to Cefalu by train or bus on your own. You do not want a car in Palermo.
Or do the reverse and add in your stop in Ragusa. Fly into Palermo ( base 1) where you don't need the car. Spend your time there in the city and with a daytrip or two as mentioned above. Train or bus to Agrigento ( Base 2) Visit the ruins. Then rent a car to drive yourselves from Agrigento to Villa Romana and go as far as Ragusa ( Base 3) Stay to visit Ragusa. Keeping your car drive to Siracusa (Base 4) possibly visiting Noto on the way. Stay in Siracusa/ Ortigia and turn in the car. Take bus or train to Catania. Finish trip with last base in Catania with a daytrip to Taormina and Etna if interested, Allocate the number of nights in each base that best serves your needs.
You mentioned you like down time and beach views. If you can add some time on to your Palermo area, stay a night or two in Castellammare del Golfo or Scopello ( both on coast not far from Palermo).