Him being adopted doesn't mean he isn't an exemplary dwarf.

Carrot left the mines at 16, when he couldn't grow much of a beard. His early days on the watch with Colin the Traditionalist would have required being clean shaven. By the time Cuddy joined Carrot likely was set and comfortable being clean shaven. As to why he continues to shave, you'd have to ask him (or possibly Angua)

However! That doesn't make him D'rzka any more than being human by birth would. He can properly ha'lk his g'rakha, and is, in every way that matters, a dwarf.

(Ask me more about the discworld) ^•^