Not sure if this helps, but Adair County, OKLAHOMA, on the ILLINOIS RIVER is found often at google. Might they have meant Adair County in Oklahoma, Illinois River area? It is in the Illinois River Basin area of Oklahoma.

There is also Adair County, MISSOURI, a state which borders Illinois and though the county is well inside the state border today, might it not have been, early on? The MO area was opened to settlement in 1812, prior to your time period.

Most of the Adair counties I know of were named for Kentucky Governor John Adair. But the Oklahoma one was apparently named for the Cherokee Indian Adair family. Don't know if that somehow connects to your search.