So many places to go.....
Good you got some guidebooks to help with planning. So many places to go......
Brenta has the best hut to hut (HTH) with VF every day. But again so many places to go with HTH with trekking and VF options. The Sella group and vicinity is another good choice.
I have done many HTH multi week trips with VF in Dolomites and since so many options, not easy to give just one recommendation.
Consider joining (if not already a member of one) a mountain club such as CAI ( https://cai-pnw.org ) for the hut discounts, insurance, and of course support to trail and hut maint.
Late June can be free of snow issues (but Brenta still has some easy glacier). however that is start of very busy season with crowded huts. Best time to go is late Aug thru mid Sept.