I've spent some time in Lucca and it fits the bill but it is firmly on the tourist map, but nothing like Siena yet. It has the historic center but is a focus for the 80-90K people that live in the area so it has more than its share of good restaurants and shopping. It is light on sites - aside from many churches - but is an 1 hourish away from Florence and 30 minutes from Pisa by train. As for local lore there is a ton - between the patron saint of the town lying visible in one of the churches to the cross carved by Nicodemus himself that magically came to Lucca and its teleportation is recreated every year in a solemn candle lit passage.
I would aim to be there on shoulder seasons if you want a more Italian experience. Also check the weather because August and early September can be sweltering. There are several non-AirBNB sites that list longer apartment rentals if you end up looking seriously at staying there.
If you really want to get off the tourist track there are many cities that might fit the bill. You could look into Pistoia which is closer to Florence on the train and is often also considered by people who move to Lucca. Prato also fist that bill but has harder times since the wool industry has declined and is somewhat less "on the up" than Pistoia. Grosseto another historic Tuscan that is firmly on my radar but I haven't made it there yet.
All that being said, I have to ask "How is your Italian?" English drops off quickly once you get outside of more touristed areas especially among the older population. Italians are vey open to working with you and I don't think you'll starve or be unable to buy a coffee. But the inability to grasp complex answers or more subtle differences could but a damper on your integration.
Hope that helps, have a great trip,
=Tod