Senigallia would be perfect. It’s actually in Le Marche (just north of Puglia), but has a huge, wide, long stretch of sandy beach with a very shallow entry to the sea — no big waves. There are hotels along the beach with playgrounds on the sand for children. And restaurants where you can sit at a table on the sand with a cool drink while watching your children safely play a few feet away. La Tartana was our favorite.
We were there in late September a few years ago and there were no crowds. It had a charming little downtown area, lots of great restaurants including two with Michelin stars. Market day on Friday. A fortress by the sea. There’s a lovely little hill town, Corinaldo, nearby — with three medieval gates and 14th century walls, it would be fun to explore. And if it’s still there, a cafe downtown called La Meridiana famous for its hot chocolate.
It’s well situated for day trips but you might enjoy it enough to just stay put for a week.