There is a public boat launch across from the Venice Yacht Club near the Venice inlet and another at Ken Thompson Park in Sarasota. We sail year round here with water temps in the mid to high 60s in winter. The Venice launch gets you into the Gulf pretty quickly with no bridges. Ken Thompson gets you into Big Sarasota Bay which is a great sailing venue, or you can go right out New Pass (draw-bridge) or Big Pass (65 ft fixed over the bay, no bridge at the pass). There are other public launches but they all launch onto the inter-coastal in places that require dealing with bridges and other hassles. They're basically there for power boats.
These places are all near the beach so the camping is going to be the hard part. Beach property around here usually means multi-million dollar high rise condos.
Oscar Scherer State Park is a great place to camp affordably (I'm assuming tent or pop-up). It's about 5-6 miles from Venice and a little under 20 to Ken Thompson.
You could also look up in Manatee County. There are several launches on the Manatee River which is quite wide there and takes you out into the mouth of Tampa Bay. There are probably camping options closer there.
Keep in mind if you're sailing in and out of passes you need to keep the tides in mind. You could find yourself fighting a 6 mph current with a 5 mph breeze. Been there; Done that.
The ox is slow but the earth is patient