As others have suggested Lucca could be a good choice but also there is Arezzo which is sort of the step-sister to Siena would be interesting and more authentically "Italian" than some other places more on the tourist map. There is no "living like locals" on the AC since there is only a crushing tourist season and then shut down.
Virtually any seaside destination in Italy will be busy in the summer since people take to the seaside for days or weeks at a time. The Ligurian coast north of CT has a bunch of interesting towns - Chiavari, Sestri Levante, Moneglia - and you'll be among tourists but I bet they will primarily be Italian tourists.
If you like the coast Salerno might be an interesting choice. I think of it because it is part of Campagna and close to Naples it has the most "land based" menus of any seaside place I've been in Italy. Anything with a tomato sauce is crazy good. There are many good seafood restaurants but also many land or "both" menus in play. Salerno is surrounded by interesting things to see - including the AC - is you want to add some day trips to your "local" experiences.
For background I spent a month in Lucca in 2024 and here are some thoughts:
Good:
- Lucca (the old town) has way more good restaurants than it should for the size. I call Lucca 'cultivated' Italian life because it collects the best of the surrounding area. (There are no muffler shops in the old city.)
- It does have all the things you want - bakeries, fruit and veg stores, weekly markets.
- There are multiple free or cheap music events every night in September.
Bad:
- It will be hot. Traditionally near the end of September the weather breaks and evenings start to be cooler. Living like a local will mean getting out early and retiring for siesta for the hottest part of the day and re-emerging at night.
- It is touristy so there are no deals to be had and you will pay retail for your lodging.
- The center is clogged with tourists during the week and while not like Siena you will hear more English (American and British) and German than you may want.
Be sure you get a place with a kitchen, AC is probably required by anyone mortal anywhere in Italy in September, how much Italian you speak may determine how much like a local you actually live. The further you get from touristed areas the amount of English spoken drops off dramatically.
Whatever you decide - have a great trip,
=Tod