It really depends on which island you are considering moving to as well as what you mean by "fine dining".
On Oahu, in Honolulu, I am sure that you will find a number of decent restaurants--probably too many to list here--that serve good dishes and wines.
But, chances are, a lot of these restaurants will trend toward new age techniques--I refer to them as "club restaurants." You know....they look and feel like a Las Vegas night club, and come with all the scantily-clad, plastic blondes to match with the food looking more creative than tasting great. In other words, they are geared toward the tourists and party crowds.
To find some hidden treasures, it may be cliched, but Zagat is a good resource and you may need to look behyond Waikiki and Honolulu.
Now, as for the other islands, it may be a bit easier to find those hidden treasures. But they will not be as abundant.
My favorite on the big island is a french restaurant in the Kona area called "Le Bourgogne." This IS classic dishes and techniques at its finest, with a pretty good wine selection. You will find foie gras, rabbit, brie, along with each meat dish being served with an appropriate sauce, and fresh artisinal breads. It is a small cozy restaurant that won't disappoint.
But remember, this is Hawaii, so do expect a little more casual atmosphere even at the finer restaurants.
On Kauai, I don't think I have found a good classical place, BUT that is probably because on that island, I stick to the local food (Tip Top Cafe, anyone?) Kauai does have the trendy "Roy's". It's good food, but I prefer the less trendy places that focus on old school dishes and techniques. And, you will always find excellent restaurants at country clubs and the major resort hotels--try Princeville on Kauai.
Those are just a few of my suggestions. We can chat more if you need! :chef: