Advice for October Honeymoon in Sicily?

Hi community, my fiancée and I are getting married in mid-October and want to plan a 12 day trip to Sicily for our honeymoon. She toured Sicily once before when she was younger with a youth music group and has wanted to go back. She loved the cities that she visited, but they were all the very common "touristy" places like Palermo and Taormina, along with a few places on mainland Italy.

We are in the beginning stages of planning our trip, with initial plans to stay in Catania, Ortigia, Taormina, and Cefalù for a few days each. However, after some research, I'm worried that these may not be right for us and I'd like some advice.

Our budget isn't huge, but we're willing to spend on experiences/stays that are unmissable. We're hoping to check a few boxes:

\- Be able to enjoy the whole trip without renting a car. We are fit and have no problem with walking around town or taking public transportation between cities.

\- Go swimming/Stay on the coast.

\- Avoid areas mainly catered to tourists, and instead enjoy local food and culture. (This is the biggest worry.)

\- Relax and take in the good vibes that we think Sicily can offer.

Can anyone suggest cities/towns that can kinda fit this theme? Coastal, with plenty of local cafés and restaurants that aren't tourist traps but still offer good accommodations and things to see and do? Huge bonus points if its somewhere with great amaro that we can enjoy throughout the trip. We appreciate any and all advice!