Disclaimer, just tossing random area stuff out there in no particular order and without grammar checking. Always check hours before you go. A lot of the large towns put info on their Chamber of Commerce FB page. Western White Mountains CoC also has FB.
Near Carroll, I can't think of anything in a 'can't miss' category. Lots of great hiking around. Better during the week so as to find easier parking/fewer crowds.
Littleton:
"cute" walkable downtown-look on the map for their 'free parking' before you go.
Worlds longest candy counter at Chutters would be a hit for kids
Good food at Schillings and Freehouse. Little Grille and Al Burritos not too shabby as well.
Lancaster: Good beer and food at Copper Pig. Mostly handhelds, but check their specials on FB. Someone on reddit reported a couple of flat beers from them, but I have never seen it. I normally drink their lager or red while I'm eating and take home a growler of stout or porter.
Bethlehem: a couple of reported by locals to be good restaurants, but I have never been. Rek-Lis and Rosa Flamingos.
Bunch of covered bridges all over the place.
Going further south max distance an hour:
Lincoln/Woodstock should have the Ice Castles open finally. It's neat if you like that sort of stuff. There is an entrance fee---just went and reread your August timeframe, so this obviously wouldn't work but I'll leave it up anyways.
There is a water park in Lincoln, but that's a waste for just a short trip and the park is pretty small.
Try not to get abducted by aliens if you go through.
Campton: Covered Bridge restaurant
Holderness: Walter's Basin. Right there on Little Squam. I try to hit them up a couple times a month. Conveniently has a gas station nearby which often has the cheapest gas in the region(Golden Pond Country Store).
Put on the tourist hat and drive east/then south from Carroll though North Conway then up over the Kanc to Lincoln. Bring a camera and take great pics from the many pulloffs. Another 'walkable' downtown in North Conway. Eat in either North Conway or Lincoln. I like Barley & Salt in NC(great beer selection and tasty fish & chips), but it is pricey and always noisy when I go through.
Be sure to hit up the liquor stores and spend money