You'll have to check the park website or ReserveAmerica for site details. But you should have no problems with a Class B. The sites further back in the park, to the east I'm guessing are directly on the creek that flows along one edge of the park. Try for one of those.
Weather in June should be for afternoon thunderstorms possible every day. Most of the sites are shaded and to a Florida resident, anyway are nice and cool. The big crowds show up for the winter months not the summer.
For beach access there are several roads connecting to the beach. The park itself is semi-urban (tho it doesn't feel that way inside the park) and there are shopping centers whichever way you turn on Hwy 41. The one to the north is better and has a nice theater too. There are all sorts of restaurants in the area.
Myakka River State Park is a place you may want to visit, especially if you want to see gators. The Ringling Circus museum is worth a visit too.