Always, always need sunscreen, especially in Florida. In FL, often what looks like a cloudy day will have a UV index of 7, as high as a sunny day up north.
Unless you are not leaving the house, you should always have sunscreen on your skin. We love Neutrogena's Helioplex, completely non-greasy and great sun protection. It's the only sunscreen I can get DH to wear.
Enjoy!