It's difficult to actually find Bulgarian culture in the resorts because they are purpose build for tourism and no one actually lives there, although St Vlas and Nessebur might be exceptions.
I would recommend being near to Varna with its museum and Roman baths, and it is major city where Bulgarians do live and work. You can also take advantage of some of the excursions, one in particular is to Veliko Turnovo, the ancient capital of Bulgaria.
A resort which is relatively refined is St Konstantin & St Elena, in older tourist guides it is Drouzhba meaning friendship. This was the first Black Sea resort dating from about 1947, and is in a woodland setting.
Peter