harting, you plan to be in Greece during the Height of High Season... the biggest Greek holiday of the year, next to Easter is Aug. 15. Furthermore, most Greeks annual vacation is the MONTH of August, so all Greeks also will be flooding to the islands. It will be crowded.
Rather than count on an online forum to do your research for you, I think you need to send your 16 & 20-year-old to the library to get an armful of Guide books and advise you ... and also have someone spend serious time on a basic website such as Greek Travel -- http://www.greektravel.com/ (BTW I am continually surprised by how many people will lay out $$$ for tickets etc for a lengthy trip, but not spend $8-10 on a used copy of say, Rough Guide to Greece or Lonely Planet Greece).
Do you have a map of greece yet? Here's one - https://www.explorecrete.com/crete-maps/images/greece.gif - (click to enlarge) that shows all islands... although some are far enough East that they must be shown in insets. If you seriously hope to find a LESS crowded island at peak season, you need to go far afield, to a place like Chios or Limnos, Nisyros or Tilos or SO far east (of Rhodes) that it's off the map totally - Astypalea. Within the popular Cyclades group, the only places that will be less crowded may be Sikinos or Kimolos. All these are less crowded mainly because they have less to offer. Also, most have no domestic flight service, and less frequent & loooonger ferry trips.
When you've selected some destinations and how to get there, we can contribute our experience to help your choose among them. but if we are to find everything for you and work out the logistics for you... we would not be voluntary advisors, we would be travel agents, and we're not. So, please come to us with some findings, and we'll be happy to help you sort them out. And do put those teens to work on this -- then they'll be invested in the trip, and everyone will benefit.
Otherwise, since time is SO short, my advice would be to seek out a good travel agency based IN Greece -- this is advised because THEY will know much more of the situation than an agent in the US who only knows the most touricized islands covered by tours or cruise ships. Also, if there's a "glitch" in arrangements, they're right on the spot to help. When you make all your local bookings for travel and accommodations, there is no added fee. There are 2 that are highly recommended as working well online with North Americans -- Fantasy Travel and Dolphin-Hellas. If you decide to contact them, be VERY SPECIFIC -- including your exact flight arrival departure times, your "wish list" of places (or idea of places), your lodging needs AND per-night budget. Good luck!!