While it is true that Fishman Karate mixes both water and taijutsu for advanced water techniques, you missed a vital part. That being the propagation of shockwaves through water molecules scattered through the air. Which can't be seen by the naked eye. Further more, a couple of more trats can be made use of in combat.
The shockwaves naturally have a highly destructive power, that much is certain. Gyojin produce them through water molecules spread out through the air which also serves a medium to transport said shockwaves into farther distances, however at the same time, passing through the water molecules increases the damage of the shockwave greatly, thus humans and other beings made of water take high levels of damage from them. So in a sense, while the shockwave do not rely on water for strength, it does indeed matter when it comes to increasing it's damange. Because of this, if you shoot a shockwave through a water bullet located infront of you, the shockwave will gain destructive force. But overall it is highly possible that the unstable vibrations of the water molecules have a high role to play in the shockwaves.
The fact that we breath and swim underwater is an example of our combat skills, even while underwater. We still swim faster than any humans, akin to a transformed Kisame, and can go on forever without running out of breath. Further more, because Fishman Karate relies on water to propagate and increase the shockwaves, using FK underwater would make our techniques just as effective and in some cases also increase their power, depending on the technique. We can also, perform regular taijutsu underwater, as our mobility is not hindered, we are after all, fishmen. Although, keep in mind that using a taijutsu technique which uses fire would result in complete failure underwater, logic and reason still applies down there after all.
FK relies more than anything on shockwave enhancements, attacks, and defenses. Keeping that in mind, a shockwave doesn't have a chart in which we can know how it interacts with elements. Thus, it all depends on the technique used, however a regular FK Shockwave will most likely phase through elements like water, or otherwise be stopped by solid elements such as earth. The full interaction of elements and FK is not clear, as not even in OP has it happened, however i'll do my best to come into reason with this and let you know. Do know that there are FK techniques that can counter elemental projectiles with shockwaves though, so with that, we atleast know that shockwaves can come into contact and even repel ethereal or solid elemental techniques.