looking for drama about orphan guy/girl looking for biological parents

I would recommend Healer, both main leads are orphans, though only one of them actively looks for connections to his family. Also in Pinocchio the main male lead searches for his brother.

Both of these have orphans finding their families:
Lady& Liar
Pan Ruk Pan Rai

Smile Dong Hae- The male lead is looking for his father. Also, the male lead's mother was adopted so they're also looking for her biological parents (his grandparents).

Golden Bride- She has a Vietnamese mother that raised her, she goes to Korea to find her father.