Not too long ago, I would have suggested you buy a good DSLR and a macro lens. These days, I would suggest that you look into mirrorless cameras, preferably ones that allow interchangeable lenses. Macro lenses generally are not cheap, but you can get a mirrorless camera body relatively inexpensively so it should offset some of the cost. I suggest a mirrorless camera because they are incredibly powerful, light, small, and just about all of them do terrific video. In other words, it'll be a camera you'll find yourself using for much more than coral photography.
I'm not going to suggest a brand or specific camera because that's a personal choice. You have to choose the camera that feels most comfortable in your hands and for which the controls are intuitive. You are in the best place in the country for camera shopping. I'm sure there is at least one good camera store in your area. The best place to go is into NYC and spend some time at the B&H store. You'll be able to handle a wide variety of cameras and their prices are going to be as good as you'll get anywhere else. Another option is to rent cameras and try them out before buying. I don't know if B&H rents but a search for camera rentals should give you several options.
Another thing you'll need to make quality coral photos is a good tripod and a remote release. Tripods are usually easily borrowed from friends to determine what you like best.