Lemmon family from New Jersey to Franklin County PA to Knoxville, Illinois

Family tradition is that my great grandmother, Kate Lemmon was Scotch Irish. 

I am trying to connect the dots with this Lemmon family.  The information I know is James Lemmon married Sarah Wilhelm in Knox County Illinois Sept. 20, 1855.  (Source: 
on line Illinois marriages in Knox County)  My direct line relative is a middle daughter
of this union, Kate D. Lemmon.  James and Sarah (Wilhelm) Lemmon moved to Kingston, Caldwell
County, Missouri
before 1870.  He was a butcher. (Source: 1880 US Census for Kingston,
Caldwell County, Missouri)  Kate Lemmon
married E. L. “Liv” Morse in 1884 in Nettleton, Caldwell County Missouri. 
(Source:  Certificate of Marriage)  Kate had three sons, including my grandfather
Howard Holden Morse.  James and Sarah Lemmon are buried in Excelsior Springs, Missouri
at the Crown Hill Cemetery.  The inscription
on their headstone reads:  LEMMON  Sarah Jane
1833 - 1905 James McD 1831 - 1903.  (Source:  photo of grave marker) 

Elizabeth Lemmon (1788 – 1880), mother of James, is buried in the Knoxville City Cemetery.  Her grave marker includes her birth date and death date.  A picture of this grave
maker can be viewed on the www.findagrave.com web site.  Elizabeth was
living with James McD Lemmon and his wife Sarah (Wilhelm)in Knoxville IL
when the 1860 census was taken.  In 1850,
she may have been living in a Knox
County household
mis-indexed as "Lanmon."  This
household included John, age 27-occupation attorney and Malvin age 32. Living
next door was Isaiah Lemmon/Lanmon and family. I think her parents may be John
Greenawalt (third child of Philip Lorentz Greenawalt) and Regina Stoehr.  There is a marriage record for “Betsey Grenawall”
and “J Lehman” (Lemmon) in Franklin
County PA
in 1808. 

The father of James Lemmon was born in New Jersey.

 Any help will be greatly appreciated.  Thank you