There are two state parks in Kentucky which come to mind: Lake Cumberland State Park which is near Jamestown, KY and about 178 miles from Covington. We consider 178 miles a very easy trip. The other possibility is Barren River State Park near Glasgow, KY-- about 70 miles further from Covington. We have stayed at Barren River, not at Cumberland State Park. The RV Park in Barren River is not on the lakeshore. I don't know ab out Cumberland State Park, but both are on nice lakes.
While I'm at it, I will mention a couple of other possibilities. There are many good RV parks on Land Between the Lakes (about 290 miles from Covington). We have stayed at Kentucky Dam Village State Park, but there are many other private and public campgrounds in this area.
A bit further away is Pin Oak Campground in the Natchez Trace State Park in Tennessee. There are campsites on the lake in that park. It would be about 370 miles from Covington. We have stayed in this nice state park in Tennessee.