My family is also at the mercy of school schedules, so we're also going to Italy in July. It's hot, but not terrible. Totally worth it. I'd avoid Amalfi (never been to CT, but it sounds similarly crowded in summer). Book a few agriturismi or Airbnbs with pools so you & the kids can cool off back at home base during the pausa (siesta).
But you don't have to sacrifice visiting charming coastal towns. 11 years ago we stayed on the Cilento coast, just south of Amalfi, and it was gorgeous, authentic summer in Italy. We were practically the only tourists who weren't Italian (I think we were the only Americans - there were a few Northern Europeans). From there, we visited Paestum, Herculaneum and took a ferry ride to Positano for the day. Positano was my least favorite day because it was so crazy-crowded.
On that trip, we also spent a week in Rome and a week in Umbria. Umbria was also not terribly crowded, and lovely. We made day trips to Assisi, Lake Trasimeno & Perugia. Our kids were 5, 12 & 16 at the time.
This summer, we're going back for a month and spending a week in Rome, a few days back in Cilento b/c we loved it so, 4 days visiting family in Calabria, and about 2 weeks in Puglia. The 'kids' are all coming with us again, and are now 16, 23 & 27.
You'll have a great time!