I have several relatives who live in that area and have heard them pronounce it both ways.
This from a St. Louisan who is constantly reminded about that city's butchered french by pronouncing Gravois "GRAV-voy", not "GRAV-wah". I mention this only to remind us pronouncing names is important because the staff of All-News KXOK in 1989 the first week with an out-of-town staff made so many colloquial (mistakes the station pulled the plug less than a year later after consistent fraction point ratings.
Another one of those 50/50 pronunciation guesses. The proper French is as Scott Simon indicated with SHART-treh. However, we all know Illinoisans could never leave a good French name alone, and have shortened it. I think you'll find it both ways in that area. It's not unlike Lewistown in Fulton County. 15 years ago, we did a story asking residents how to pronounce the name of the town. The Guide says it is LOO-is-tuhn. The residents were evenly split - half said LOO-is-tuhn and the other half said LOO-is-town. I think in the case of the state park, you're right either way, Bill.