The wonderful old town of Nessebar is so steeped in history, rich in architecture and culture that it must be the perfect antidote to Sunny Beach. Its history dates back thousands of years and it has borne witness to many different civilisations (Thracians, Romans, Byzantine, Ottoman & Greek) over its millennia of history. It was declared a museum city in 1956 and inscribed in the UNESCO list of World Heritage sites in 1983.
The old town is situated on a rocky peninsula  less than 1km long and 1/2km wide and connected to the mainland by a single road with restricted car access. You will need to park in the new town. Take your time to wander the cobbled streets and explore the remnants of fortress walls from Roman times and the Middle Ages, old Byzantine and Bulgarian churches, and characteristic 19th century houses. There are plenty of cafes, restaurants and small art galleries and shops should you become overwhelmed by all the history. You should, however, not miss the archaeological museum.
Some years back Nessebar seriously jeopardised their UNESCO listing with illegal construction and other 'offences' which sadly reflected a local disregard for the historic gem on their doorstep. http://whc.unesco.org/en/list/217