In Census 1900 Charles M Johnson (age 55 b. Jul 1845, KY) is residing in Ladonia Town, Fannin, Texas with wife Nannie (age 34 b. Oct 1866, TX) and children Ruby (Age 7 b. Feb 1893, TX), Franklin (Age 5 b. Feb 1895, TX) and Vaughn (Age 3, b. Mar 1897, TX). The census indicates they have been married 8 years so about 1892. Ref: https://familysearch.org/pal:/MM9.1.1/M3GS-PK2
Charles' parents are given as both being born in Kentucky and Nannies parents are both listed as born in Missouri. Since we are trying to work backwards I'll skip the 1910 record where they are living in Alex, Grady, Oklahoma, having moved there sometime between 1897 and 1902 given the birth places of the children.
1860 Census has 1 Charles Johnson living in Bowling Green, Warren Co, Kentucky of about the right age. The household (804) consists of David Johnson age 40, Lucinda age 35, Charles age 17, Elizabeth age 15, William age 13, Margaret age 7, and Nancy age 3. http://archive.org/stream/populationschedu397unit#page/n363/mode/2up - not 7 Children, and Charles is not the youngest.
Mammoth Cave is actually in Edmonson Co. and sits close to the intersection of Edmonson, Barren, and Hart counties. Using familysearch, In 1860 Edmonson has no good matches born in the right period. There is a Charles F. Johnson enumerated in Barren, and a Charles E. in Hart - I have not examined these records closer though.