Town to visit between Rome Airport and Chianti Not Orvieto

It may be too soon to stop, but Viterbo has some sights and some charm. If you are interested in a gourmet-level prepaid lunch at a winery (with tastings at the table), consider Avignonesi. Our driver simply drank only tasting amounts, and we did our wine-shopping after lunch to delay departure. This meal was a high point of our Tuscany stay, which was sleeping outside Sinalunga.

Your proposed towns are nice, but I suppose you'd pick a place that's harder to reach from Adine.

Castiglione del Lago is a nice place, with a cooling town lake-beach, and a town castle and museum. We were told that it has good restaurants, but we did not need a meal the day we visited there. This counts as a minor historic hill-town. This would be less time-consuming than Avignonesi.

In my opinion, Pienza is perfect for the kind of short stop you want. It overlooks the Val d’Orcia, and Is a quite small example of a Renaissance planned town. Unlike towns that just grew, the architecture is harmonious. We had a tour through the papal palace with a guide from the TI office. That was years ago, so I don’t know if that’s still available.

Exit the A1 freeway at CHIUSI-CHIANCIANO TERME and visit Montepulciano and/or Pienza. The detour will be only one hour longer, plus any amount of time you plan to visit the two towns.
However, since Adine and Montepulciano/Pienza are only about one hour apart, you could probably visit both from Adine, after you are settled at your place. In that case you might consider stopping before, somewhere closer to Rome, some place that you wouldn't be able to visit from Adine. Aside from Orvieto, which you already visited, Civita di Bagnoregio is a popular destination. In that case you would exit the A1 freeway at ORTE, and proceed via regular slower two lane highway.
If you just care about lunch, then I wouldn't bother. I'd just stop at a AREA DI SERVIZIO (service area) along the freeway. Those rest stops feature grocery shopping (deli type supermarkets with lots of Italian specialites) and the restaurants of the Autogrill chain, which are actually really good. There are five Service Areas with Autogrill locations on the freeway between Rome and Siena.
Adine is a very small hamlet, with no more than 10 houses, and I doubt there are any shops. I know there are supermarkets of the COOP chain in Gaiole and Radda, but that would be a bit north of Adine and you are coming from the south. If you need to do major shopping, I would stop in Sinalunga, along the way. There is a large shopping center with a COOP supermarket there and it's close to the freeway. Location displayed below:
https://goo.gl/maps/qxpqeXC4jLDb7JFm8
Sinalunga is less than one hour away from your final destination in Adine.
Once you are settled in Adine, the shopping closest to you is probably Radda. They have a COOP supermarket, although not as large as the one in Sinalunga, but it's closer to Adine (10-15 min drive north). Mark this as your favorite location as well:
https://goo.gl/maps/KMsTsJfpk3NjfkZi7

I second Pienza. It's very easy to find parking and it's flat. There is a lovely wine shop Di Ghino, they have an incredible selection of wines where you can pick out a bottle for your first night in Italy. There is also a wonderful gelateria, Buon Gusto. I love Montepulciano spent almost a week at this location; it's a steep uphill climb from the parking and has more territory to cover than Pienza. I feel you would need more time and a lot of energy to take on this city after an international flight.

Montepulciano and montalcino are both beautiful places with great views, food, and wine.