There was one major battle in Florida. It happened in Olustee about 40 miles west of Jacksonville. There were smaller engagements, but that was the only significant one. Most of the action in the state was blockade running. There was quite a bit out of the St. Johns river. A Union transport, the 'Maple Leaf' struck a torpedo (mine) in 1864 and sunk in the river. It was never salvaged, but moved by the Army Corps of Engineers and is still there today in about 20 feet of water. It is a national historical landmark so you cannot dive on it plus the river is muddy and dangerous in the area. The artifacts that were recovered, along with parts of the ship are in the Jacksonville Museum of Science and History.
The button is Navy and looks similar to NA112 in Alpheus Albert's book "Record of American Uniform and Historical Buttons" which along with Daniel Binder's update book are considered the bibles of Civil War button collecting. It is a little cruder and may be a contract or C.S. out of England. You may want to contact William Leigh at
www.civilwarbuttons.com. He is one of the more knowledgeable on these.