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Is there a way to make it so that your unit cannot take damage underwater, but he can also still shoot his weapon underwater?

Nope. Sorry. There are ways you can make your map look like it's underwater without it actually being underwater, take a look at Xavious's Gylesh seabed or Mav's first rebel ops map. (I forget the name)

Aqualaris Deep Sea (Mav's Map) imitates being underwater well

Ahh, alright! Is there a tutorial I could look up?

A dedicated tutorial on how to make a map that looks and feels like it takes place underwater? A tutorial on being creative? No.

What you can first do is decide what kind of underwater environment (e.g. crystal clear or dark and murky) you want your map to have and compile a list of the qualities that that environment has. Think about how you can translate those qualities into a map using world textures, lighting, effects, sky parameters, etc. If haven't done so already, play the two aforementioned maps to look for the characteristics that give them the underwater appearance without actually using the real game water. Then ask yourself what methods were used to achieve those characteristics. If don't know how to get a certain feature to work (e.g. the shadows on the ocean floor like in Mav's map, or the limited visibility in distance in Xavious's map), then feel free to ask about it. This is going to require experimentation on your part.