I cannot be certain of this, because it's a very hard test to make, but based on my knowledge I would say that yes, you do need to be wearing one of the helmets, since:
- It is known that for the Firewalker Achievement you need to both equip the Firewalker Boots and Walk over Fire Fields. Either of them alone is not enough to progress in the achievement. Using that logic, for Deep Sea Diver we could also assume you need both the helmet and the special tiles.
- It's been known for some time that one way to detect if a specific floor counts for the achievement or not is to test with the helmet. If you get drowning damage without the helmet, then that floor counts. With that in mind we can speculate that the exact trigger for the achievement is the helmet protecting the character from the drown damage.
As for your specific case: even if you were using the helmet, or if using it is not necessary, your numbers aren't extremely high in terms of how long you need to get this achievement. In my experience it takes between 80 and 100 hours or non-stop walking to get the achievement (by e.g. holding to keyboard keys).
When you're hunting, however, you spend most of your time standing. How much is very hard to tell, but I would say that you walk at most 50% if what you would have if you were just walking. Using that logic and the numbers you provided, you'd have 79 hours of underwater walking. That's a lot and close to my estimate above, but nothing that would make me say you should absolutely have the achievement already even if you were using the helmet at all times.
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