Quieter location to stay for 3 weeks

I doubt you'll find much available for this year. Europe , and Southern Italy, is crowded, well into the fall. Probably best to go later, or start planning for next year. Lots of pent- up post- COVID travelers this year.
Safe travels!

A friend of ours just returned from two weeks in Lecce. He and his family rented a small villa (Airbnb) and I believe it was @ $250/day. (Total of six people). Very economical. They did have a car.
They just loved everything about the area. Off the beaten path and no other tourists…..A bit difficult to get there but well worth it.

Puglia is an area not visited by many American tourists altough the Italians are well aware of it. Just make sure the place you decide is not in any Rick Steves guidebook.

I've been to Lecce and I agree it's a lovely place. Not far from either coast.

Will you have a car or are you relying on public transportation?

Another Lecce fan here, but I'd take a look at actual, historical, day-by-day temperatures for any southern location before planning a long stay there beginning in mid-September. It could still be quite hot at that time of year.

Timeanddate.com -- September 2022 weather in Brindisi

Unfortunately, my go-to website for this purpose doesn't have data for Lecce for last September, so I have to show you Brindisi. Brindisi's a bit farther north and on the coast so it may tend to be cooler there.

In any case, it's important when considering weather to check several recent years rather than just one. You can change the month and year displayed by using the pull-down box at the right, above the graph.

Puglia is enormously popular with Italian tourists as well as with other nationalities so I'm surprised dianae99's friends didn't see any other tourists, perhaps not American tourists.....
Having said that, the peak season is July and August for the sea. June is less hectic and things really tail off in September after about the 10th when schools in the north of Italy start up. I usually visit Puglia last minute from mid-September and have not had problems with finding accommodation. Although I look for apartments and finding one place for three weeks could be difficult last minute.
I've read the reports of there being a lot more tourists in Italy this year, but I get the impression this is the situation in the big tourist destinations like Rome, Florence etc.
In other areas, Lake Iseo where I rent an apartment and the mountains where I live, for example, there are a lot fewer requests than for last year.
Just my impression but I think people were scared off big cities by Covid until this year. And a lot more Americans are traveling this year, too and many of those concentrate on the main sights.
As a place to stay two or three weeks Trani gets my vote, lots of places to day trip to but nice vibes in itself:
https://slowtravelitalyspain.blogspot.com/2023/04/why-i-love-trani.html?m=1
But anywhere in Sicily would work well too.
Ortigia (Siracusa) or Trapani.
Trapani is another of my favourite places..
https://slowtravelitalyspain.blogspot.com/2019/03/trapani-october-2016-trip-report.html?m=1

Abruzzo would be ideal in September...Vasto, Lanciano, Ortona are bigger but lovely towns near the coast; Casalbordino, Pacentro, Caramanico Terme, Guardiagrele are beautiful smaller mountain towns but on the Adriatic side so an hour or less from the coast. The roads are good so you can explore coasts and mountains. The food and wine are outstanding. Lots of harvest festivals and small wineries to visit too.

The other side of the Apennines is chock full of small atmospheric towns but not as close to either coast.

In any case, there will not be lots of tourists around. Right now, it's only crowded on the beaches, lots of room up here in the hills.

I'd look at smaller towns in Puglia like Nardo, Ugento, Galatina.

I haven't yet been to Abruzzo but have Sulmona on my wish list. I would second Trani. It's one place to which I'd love to return. My problem with the tinier towns is that there is not as much restaurant choice. Since you mentioned cuisine, that's a consideration.

100% Abruzzo. Weather will be good. You are close to amazing mountains (La Majella) and the coast. There will not be many tourists. I spend most of my quiet vacations there as I have family in the region. There are many little cities that are wonderful as well as traditional villages worth a visit.

You will really benefit from having a car, and, depending on where you go there, you may not find a lot of English speakers. You will find the people honest, welcoming and helpful. You will not find bad restaurants as easily as you do in the big touristy locations.

There are many cultural places to visit - historical, art/design/museums, nature, food producers/vineyards/oil producers. It may sound cliché but there is something for everyone here.

Another vote for Lecce. We stayed there in June of 2022 (after a RS Trip), at a wonderful B&B, which I recommend: Roof Barocco Suite B&B, Piazzetta Arte della Stampa 13, Lecce - 345 0021403 - 345 0021403 - 345 0021403. Check it out, we loved it and it has wonderful reviews.
If you have a car, day trips to Otranto, Taranto, Gallipoli, Ostuni, Martina Franca, to name a few beautiful surrounding towns, are the perfect for 10 days in S. Puglia.
Maybe 10 more days in Bari and nearby towns. My 2 cents.