I will second Cape San Blas. We stayed a week at the state park out at the end of the peninsula, the public access beach that is dog friendly is just outside the gates.
Also, in Venice Fl is an off leash beach dog park:
http://www.yelp.com/biz/brohard-paw-park-and-beach-venice
We met some nice people there.
Also, in Venice Fl is an off leash beach dog park:
http://www.yelp.com/biz/brohard-paw-park-and-beach-venice
We met some nice people there.