New unemployment claims in Florida tumbled significantly to finish off May. There were 5,167 first-time filings in the state for the week ending May 30, according to figures provided by the U.S. Department of Labor (DOL). That’s a fall of 803 claims from the week ending May 23, when there were 5,970. It’s the third week in a row that initial filings fell in Florida. The sharp drop in new claims comes after a turbulent May for the Sunshine State.…