Hi Charles- I'm not related, but will try to help.
Edward Marshall married Mary Crawford 12/18/1838 Muskingum Co. Ohio by Harvey Camp, M E Minister.
1850 Rich Hill, Muskingum, OH Marshal, Edward 32 PA Mary 29 OH Sarah A. 9 OH Mary J. 8 OH John W. 5 OH James A. 4 OH William L. 1 OH
According to findagrave there is a Lazarus Marshall d 6/22/1854 age 67 buried at Oakland Methodist Church Cemetery, Muskingum Co., OH
These clues put the families somewhere in the area east of the Blue Rock State Forest.
There is a John Crawford (b. 1806 IRE) family 3 doors from Ed Marshall.I suspect this may be a brother to Mary rather than her father, as he has a daughter named Mary who is 19 in 1850.There are lots of Crawfords born in Ireland in this area.Most of the patriarchs are dead by the 1850 census though.
Early records (pre-1850) are scarce.Often land or church records may provide the only clues.County Recorders often have old land records (Grantor and Grantee) which can help connect families.Ed Marshall does not show that he owned land in 1850, however, the John Crawford on the same page did.Early wills can also be helpful, if they were filed and survive.
With Lazarus in a Meth Cem & Ed & Mary married by a Meth Minister, it looks like the families were Methodist.I think there is still a Methodist Church in Chandlersville that might have old records or know of the early records.Harvey Camp appears to have been a circuit minister of the area in the 1830s, so "church" records may not exist.
You can search census records etc @ familysearch.The Muskingum Co Genweb site may also be helpful.Both are free.
Your search is likely to be similar to putting together a jigsaw puzzle with several pieces missing.Good Luck!