According to the 1850 census for Marshall County, Mississippi, James Monroe and William Edmond Puryear were born in Tennessee. Their brother Walter Randall was born in Alabama and brother Thomas Jefferson was born in Mississippi. Brother John was born after the 1850 census in Mississippi.
I have been unable to find out what happened to William and Mary Pitt Puryear after the 1850 census.
Edmond R. Pitt served in the War of 1812. His name appears on the 1812 census for the North Carolina Militia, Wake County Regiment, Seventh Company.
The Edmond R. Pitt family is in the 1820 census for Chatham County, North Carolina. The family is in the 1830 and 1840 census for Franklin County, Tennessee. The family is in the 1850 census for Lamar County, Texas. You have the 1860 census info.