Mangalia
Mangalia, known as “Callatis” by ancient Greeks, is a little town that less about the beach and more about its minor but worthwhile archaeological sites (and health centers for elderly European tour groups if that’s your thing). Mangalia is Romania’s second-most important harbor, though mainly for military purposes and ship repairing.
The attractions here will only take a few hours to tour and are hence a nice break from the beach when you scorch your forehead one too many times.