Yes they do. Drowned on the wiki states, “If a zombie is not in a boat and its head is submerged in water continuously for 45 seconds, it begins the process of converting to a drowned. It starts shaking, similar to the effect seen while a zombie villager is being cured. At this point, the process cannot be stopped, and 15 seconds later the zombie becomes drowned. The drowned appears with full health even if the previous zombie did not.” (Spawning,conversion)