Most fruit trees that are bought from professional sellers are grafted, which means that a hardy root stock was attached to a fruit bearing top half. If your new growth is appearing from below the original graft, it will not give you Florida King peaches no matter how long you wait, because it has the genetics of the hardy root stock. Your new growth might give you a less tasty peach eventually and it might not, no way for me to tell from here. A picture would help us to know if your top growth is really dead or if patience would be helpful for you. The bottom growth is very likely to be useless to you. They are called suckers and everyone usually cuts them off each year.

Most nurseries will replace trees that fail to break dormancy.