By itself 3D does not really matter
Here is the thing: your problem will be to make combat any different, if you want a thematic campaign. Introducing the 3rd dimension will not significanty change how D&D combat plays out.
Why? Because the combat environment is nothing but the information of where characters are and where terrain features are. What do players ask the GM about? Can I reach that cover with only my move? Can I get those monsters all in my AoE? It is only about the pairwise distance of things. And you can already be 10 feet from one enemy, 30 from another and 50 from a piece of cover on a 2D map.
What the 3rd dimension will do is that now not only 3, but 4 things can be equidistant from each other, which is pretty much inconsequential for us. It is possible that things that provide substantial cover have to be bigger to do the same job, but most area control AoE-s are already fairly large, and in the case of terrain features, you can literally make them as large as needed. Another effect will be that now more creatures can surround a single creature. But on a 2D map you can already be surrounded by 8 creatures (medium vs. medium on a square grid), and in most combats there will not even be 8+ combatants on one side. Even in "mass battles" with two dozen monsters, I have never once saw a situation where 8+ enemies wanted to surround a single character.
So we can see that there is no significant geometrical effect. Besides the rules of underwater combat outlined in the books (which will likely be mitigated or completely avoided by any party worth their salt or any monster actually living there), there are really only two things that need attention.
Most things sink.
Dropped your sword? Sinks. Was swimming and speed dropped to 0? Sinking. Threw a dagger? Will sink. Depending on how likely it is that you will have the chance to go after things that sink, the attention given to these situations will change.
The most common reason people drop their weapons in D&D 5e is to switch to another and still be able to attack with it on the same turn. In a usual environment you can expect to be able to retrieve it, but that will change here. This will lead to people rarely switching weapons in a combat situation, as it becomes too costly in either actions or money. The switch is usually from a melee weapon to a ranged one or vice versa. Thus characters will be more dedicated to one role or the other.
You are also usually assumed to be able to retrieve thrown weapons if you didn't have to leave the combat site (eg. to retreat or catch a fleeing enemy). Daggers, handaxes and throwing stars will sink after one use, while a javelin might float due to its wooden shaft. Either way, the retrieval of the items will be much more problematic, which will discourage their use.
The above two can be avoided (and they automatically are by most spellcasters), but in combat characters will go unconscious. If you are swimming by your own power, you will sink when you cannot move. So a downed teammate is not only rolling death saves, but also getting farther and farther away from help. This can make it a truly pressing issue, which is not always how it feels in land combat ("I still have time to kill this orc, I can save him next turn all the same."). The lethality of combat might rise a bit.
These are not issues that necessarily need to be "solved", though. Leaving them as is will not break anything, and might even lend a bit of unique feel to the environment.
On land it is easy to introduce terrain features to combat. Most places have trees or shrubs, and even in a sand desert there are dunes that can provide cover, for example. However, the deep seas are fairly empty, as not many things float at just a given depth apart from fish. A fight on a featureless plane (now in 3D!!!) is alright once in a while, but you will need to come up with a lot of ideas if you plan to run a whole campaign underwater. You could run a combat at the bottom, where there could be bentic flora, but then you are practically running a land fight, so if you want a heavy theme, I would not recommend relying on it too much. I will list two ideas I could come up with quickly, but this is far from enough for a campaign. This is an issue you have to be prepared to tackle.
- Beside the huge coral reef. They are exotic and cool looking and can really form mountains even in our non-fantasy world. You can take cover inside it or on the other side of it.
- Hunt for the Red October. There is a new gnomish contraption (a submarine) and the party is tasked with getting it. The submarine can be fairly large to serve as terrain and can float. It can also move, which can have interesting consequences.
You mention that there are spells that rely on there being ground or air around. Well, there are also spells that only work when there is plenty water around, like control water. Whatever the place, there will be circumstantial abilities that do not work to their full extent or at all.
As your players are aware of the theme of the campaign, they can make informed choices about which abilites they want to take. I would recommend against changing abilities or spells for ease of use. One, this might place the focus on rarely used choices, making the experience different. Two, if you really want to, you can introduce these as rewards ("As thanks for saving their village, the sahuagin shaman offers to teach you the spell iceball.").