Since Zoan awakening is probably a constant boost in stats, it isn't specifically mentioned. It's not even mentioned in Kaido's case. I doubt his awakening would give him new modes or abilities.
@sanjikun well, Doffy stitching his organs together is a trick too, but you give him credit for that. Even without the flame, King is pretty durable; overpowering Zoro's tiger hunt and only having his mask partially shaven off, while not carrying his flames. And the flame defense is a legit defense. He can also fight while having his flames on; in this case he is just slower, while without flames, he is incredibly fast.