In Clarksville, this state park has only rapids, no falls, but it is of interest for its 390-million-year-old fossil beds, which are ideally seen in late summer/early fall when river levels hover around 13ft (the river level is controlled from the Kentucky side so it can still be a crapshoot). The interpretive center explains it all. The park sits right across the river from Louisville, KY.
Falls of the Ohio State Park
Indiana