Sgt James R Dimpkins' name spelling has been reported by one source as spelled "Dimkins. All but one sources checked reported this man to be listed as James R Dimpkins. 
DOB Estimate of 1804 is based on the fact James R Dimpkins was a Sargeant by 1835, time needed for him to immigrate to America and reach San Antonio, Bexar, Texas and the other Defenders' ages of the Alamo.
He immigrated to America. We find him by 1834 in New Orleans in this reference.This U.S. and Canada, Passenger and Immigration Lists Index,
1500s-1900s about James R. Dimpkins shows:
James R. Dimpkins was present in Bexar colony, Texas by 1836.
Arrival Place: San Antonio, Texas
Source Publication Code: 1494
Primary Immigrant: Dimpkins, James R
Source Bibliography: DEFENDERS OF THE ALAMO. In Georgia Genealogical Magazine, vol. 37:2 & 3 (Spring/Summer 1997), pp. 84-88.; pg 85. 
In New Orleans, James joined the New Orleans Greys under Capt. Thomas Breece.. They marched to Texas together.
Fall of 1835, James R Dimpkins arrived in San Antonio early enough to serve in the siege of Bexar (Oct-Dec of 1835). After fighting this battle, he was promoted to Sergeant of infantry of the garrison with a different commanding officer: Capt. William Blazeby.
List of Travis men.
In 1836, Sgt. James R Dimpkins was killed in the Battle of the Alamo on March 6, 1836, with the other Defenders of the Alamo.  A memorial is in San Fernando Cathedral. 
Dickens County, Texas is named for James R Dimpkins "Dickens".