Oscar Scherer park is on the south side of Sarasota and actually in Osprey. As far as shopping, there is a Walmart about 2 miles north of park. I've camped there a few times and I personally like the upper loop, sites 1-13 along the creek or on the backside 21-35. I stayed in 3 & 5 a couple of years ago. If remember right 5 was almost directly across from the bathrooms. In most of the sites you can't see south creek from your site for all the trees and scrub brush but some sites had a path to the edge of the creek. My opinion the creek is a bit underwhelming. I would also call the park and ask questions and get the rangers opinions-nothing better than hearing it 1st hand.
There is biking/walking path called Legacy Trail which is an old railroad bed that the county has black toped located out about a 1/4 mile past Lake Osprey. I believe it runs the entire length of the county. People are always riding bikes and walking up and down the Legacy trail. It's wide open so take water.
As far as things to do, check out March 2017 issue of Trailer Life. Fort Myers is south about 1-1 1/2 where the Edison and Henry Ford winter homes/museums are. Beaches are no problem, Siesta Key, Lido beach, Sanabel Island or farther north is Longboat Key to name few.
I agree, Myakka State park is a another great park to stay in. My opinion it is not very private and I don't remember the FHU area very shady. I have also stayed there several times both winter and summer.
You could alway stay at Oscar Scherer and then spend the afternoon/evening at Myakka State Park looking/watching the gators along with deer and other wildlife and maybe take a tour of Lake Myakka on 75 passenger air boat.
Weather in June could be afternoon showers but otherwise great.
These are my only opinions but I'm sure whatever you choose will be good.