The only William I have listed with an 1813 birthdate is William David SWINT, s/o John SWINT, Jr. & Rachel SPAULDING.I have ca. 1813 for his birthdate but have no birthplace.He died 2 Feb 1887.He was married twice.His wive's names were Sarah "Sally" JOHNSON and Tabitha BAILES whom he married on 27 Feb 1852 in GA (I don't have the town or county).
John SWINT, Jr. was b. ca. 1785 in Richmond Co, GA which became Columbia County by 1790.He was 10 when his parents (John SWINT & Elizabeth CARTLEDGE) moved to Hancock Co, GA.
I have a number of Williams listed, but no William Roberts and most of the Williams I have birthdates for and they don't match except for the William David above.
I also checked Mildred Ezell's book "SWINT" and she had no William Robert listed. There were four pages with "William R." but nothing that matched with an 1813 birthdate.
Sorry I couldn't be more help.Good luck in your search.If you have any additional info that might help pin something down, I'll be glad to check again for you.