OP is missing two huge issues, I think...
1: Have you been to Europe before?
I'm guessing no, based on your post. If not, you may have a wildly inaccurate idea of the meaning of the following phrase...
2: "We love sunny weather and relaxing on beaches..."
IME, to most Americans, the idea of "relaxing on a beautiful beach" is a picture of a beach in the South Pacific, Thailand, a TV commercial of a beach in the Caribbean, maybe an actual experience in Florida or California. Nothing wrong with those experiences, but "relaxing on a beach" in Europe is nothing like any of those.
If you want a "beach vacation", IMHO you should go to someplace that has the beautiful beach of your dreams - broad swaths of fine white powdery sand, with crystal clear, warm water, and wide stretches between you and others and plenty of room to stretch out. That is generally NOT what you will find in Europe, even in places that are popular beach destinations. You will find crowds (in especially in August), you will usually find pebbles, not white-sugar fine sand, and you will often find tightly packed rows of chaise lounges (all filled with naked, sun-burned northern Europeans roasting in the sun...) rented by the hour. I dunno about you, but that's not the stuff of my beach dreams.
Now, that said, there are many beaches in Europe that are nice and great places to visit - but they are NOT like "the beach of your dreams" that pretty much every American I know envisions. I do think they're worth a look and maybe a brief stop - but IMHO they are NOT worth flying half-way around the world for.
IMHO, if you want a great "beach vacation" it's very foolish to plan a trip to Europe for that. If that's what you want, go to Cancun, Kauai or Phuket or a hundred other places that are all about the beach. You go to someplace to experience what they have that you can't find elsewhere. I go to Europe for the things the OP mentioned as "also" items: culture (mostly). History. Art. Museums. Churches. Food. Adventures? Maybe - depends on your definition of adventure, I suppose.
If you're turned off by your friends' description of how touristy the beaches in Greece are, I think you will be pretty bummed out by most of Europe's best beaches in August.
You want relaxing beach time? If you can't change your dates, then either pick another corner of the world (better be prepared to go someplace either very remote of very exclusive/expensive), or plan a different focus for your trip to Europe. Personally, given all the other great things that Europe has to offer, I'd consider it kind of a waste to spend much time on a beach there in August - like, say, going to Tahiti and spending most of your time admiring their art museums and churches (no offense to Tahitians, I'm sure their museums and churches are nice enough - just not so impressive when compared to Italy's).
Good luck.