Your family doesn't have any tie in to the Grigsby Appleton family of northern alabama.This family came from the Newberry County area of SC prior to their move to AL in the 1840's.
It seems that your line came from TN and NC and moved to Panola County, MS in the 1830's.It seems that they were in Robertson County, TN in the 1830 census and then Panola Co., MS in the 1840 census.Your specific line moved from Panola County in the 1850's when Jane Appleton, born NC was listed in the 1870 census for Johnson County AR with her two children:William J. (James) and Nancy W., age 13.By the 1880 census for Logan county, Arkansas James Appleton is head of household with his wife, Emma and his widowed mother, jane and his siter, Nancy.
So tracing them back it goes Logan County, AR 1880 census Johnson County, AR 1870 census Panola County, MS1860 census
the Panola County, MS Appletons are related to Thomas Monroe Appleton and this family seems to have been in Panola County, MS in the 1840 census and Robertson County, TN in the 1830 census.
No exactly sure how it all fits together but again, it is not tied into the Grigsby Appleton family of Alabama.
Continue good hunting. Sue Appletonjsappleton@alltel.net