I can help you some with your earlier Cullum/Cullom history.
Jesse Cullum's wife was Mary "Polly" Lovell from Halifax Co, NC. She was the daughter of William Lovell and Mary Harrison.
Jesse was definitely the son of William Cullum Sr and first wife Amy Hix/Hicks. He was named as a son in William Cullum's Halifax Co, NC will dated 18th day of June 1809, proven August, 1822.
William Cullum Sr was son of Thomas Cullum Sr and Elizabeth Johnson of Surry/Sussex Co, VA. His birth is recorded in the Albemarle Parish records. William Cullum Sr moved from Sussex Co, VA to Halifax Co, NC between 1777-1780.
The parents of Thomas Cullum Sr have never been proven, but Elizabeth Johnson was the daughter of William Johnson Sr and Phoebe -?- (suspected to be a Pettway, but no proof) of Surry Co, VA.
There is a 1797 will for a Thomas Cullum in Sussex Co, VA, but this was Thomas Cullum Jr, not Thomas Cullum Sr. No estate records for Thomas Cullum Sr have been found, but he probably died shortly after 07-20-1774 when he last appeared in the vestry records.
The Riggan family in Davidson Co are also Cullum descendants. William Cullum Sr's son Jeremiah Cullum (Jesse's full brother) left several children including a daughter Amy Cullum who married John Riggan and moved to Davidson Co, probably with her uncle Jesse Cullum. Asa H Riggan was one of the children of John Riggan and Amy Cullum.
Halifax Co records indicate that John Riggan was the guardian of Amy's siblings: Mary Cullum (has not been found in later records) Beder/Bedee/Beady Cullum m Gardner Williams 1826 Davidson Co, TN Frances "Fanny" B Cullum m John Jones 1830 Davidson Co, TN Theny Cullum m Thomas Reaves 1824 Davidson Co, TN Her name was given as Lincy in the marriage record but as Theny in the guardian records.
The Williams and Jones moved to St Francois Co, MO. I haven't located the Reaves in later records.
Another sibling, Sarah Cullum, married John's brother Robert Riggan and lived in Warren Co, NC where the Riggan family was from. Although Jeremiah Cullum's Halifax Co will did not name daughter Sarah, he did leave a greatcoat to Robert Riggan.