We stayed in Cavtat ('sav-tat') a few years ago at the Hotel Croatia which is on a rocky promontory overlooking the village. Nice modern hotel (bit plastic). Village is a quick (steepish) walk away, with nice restaurants and big yachts. But just pebble beaches.
Dubrovnik old town is now utterly crammed with tourists, made worse by a recent influx of Game of Thrones nerds, but worth a walk around (get there first thing).
We did a boat trip to Lopud island, which had nice beaches and cafes. I think the beaches are on the opposite side of the island from the cafes, hotels and harbour (maybe 15 mins away). Definitely worth investigating as a place to stay.
We found the top inch of the sea was nice and warm, and the rest freezing, but that was in June, I think.
Whole country is full of Russians, as they can almost speak the language.