I have camped at the Walter Farr Equestrian center in Baton Rouge several times. It is south of the LSU campus just off of the Ms. river levee. It does offer water and electric, it essentially has no shade but large spaces and easy movement within campground. Don't even think about a spot for an LSU football weekend. If you are into walking or biking there is a great path that you can access from the park that runs on top of the river levee for several miles.
The only negative is that you will encounter several miles of significant traffic getting to the park, best to avoid the rush hours for sure, but then a lot of people will tell you it is always rush hour in BR!
La. has some great state parks, we particularly like Fontainebleu SP outside of Mandeville, good biking and walking trails and awesome
sunrises and sunsets on Lake Pontchartrain. Farther into south La. is Palmetto island SP great place to relax and get away from it all.
Have a great trip!