Edmund was born in Philadelphia Pennsylvania October 28, 1712 and died after 1784 in Columbia County, Georgia. Edmund married Eliza Keble on December 9, 1743 in Craven County, South Carolina. Lucretia Elizabeth Keble was born October 7, 1728 in Philadelphia, Delaware County, Pennsylvania and died in 1765. Some online trees said she died in Carolina, Puerto Rico and others say Edgefield County South Carolina (which makes more sense).

Settlers of Maryland, 1679-1783 lists Edmund Cartledge in Hickory Tavern, Prince George County, Maryland in November 1738.

According to Carolina Cradle: Settlement of the Northwest Carolina Frontier Edmund III removed to Anson County North Carolina before the spring of 1745 (North Carolina Land Grants, V, 323). This has to be Edmund III as Edmund Jr and Edmund Sr. had passed away by this time. This was the second Quaker migration – heaviest between 1714-1740, based mostly on economic reason. Carolina Cradle: Settlement of the Northwest Carolina Frontier stated “As early as 1660, George Fox suggested that land be purchased in America for use as a Quaker colony, and the low rates at which land was offered in Pennsylvania after 1700 caused many Friends to act on Fox’s suggestion. The Free Society of Traders, including many Quakers, bought twenty thousand acres of and in Pennsylvania to be developed. Robert Tuner, and Irish Quaker and one of a committee of twelve directing the Society’s activities, went to Philadelphia with seventeen indentured servants as early as 1683 Settlement of the interior, which began after 1714, accelerated these early activities, leading to the establishment of land companies and the organized planting of interior settlements. Among the prominent Quaker merchants and traders who financed and advertised these speculative enterprises were Hugh Parker of Pennsylvania and Maryland, Edmund Cartledge (Note: This is probably Edmund Sr.), Alexander Ross, Edward Shippen, George Croghan and Morgan Bryan, of Pennsylvania; and Benjamin Borden of New Jersey.”

Edmund III was also listed as one of the original justice of the peace for Anson County. This same source also stated that Edmund III was a key figure in the development of the Pee Dee settlement (in the vicinity of modern Wadesborn). The State Record of North Carolina Laws, 1789-1790 Page 272 states:

An Act for appointing and laying out a town on the land of John Jenkins, on the south side of the Pee Dee river, in Anson County; and for other purposes therein mentioned. …. … that as soon as the proprietor of the said land shall acknowledge his consent and concurrence in open court, of the said county, to have such part of his said land laid out for a town, as is herein after directed, Mr. Charles Robinson, Mr. Caleb Howell, Mr. Thomas Tompkins, Mr. William Forbes, and Mr. Edmund Cartledge, are hereby nominated and appointed commissioners; and they or the majority of them, are hereby invested with full power and authority, to lay out fifty acres of land on the said plantation for a town, by the name of Gloucester, and to lay out the same fifty acres into lots of half an acre each, with convenient streets and squares, for a church churchyand and market place.

A history of Richmond County (North Carolina) states:

We learn from the May Wilson McBee Collection, Vol. `that Cartledge Creek got its name from a landowner, Edmund Cartledge, who had a land grant No 1323 September 27, 1756. In 1758 Cartledge sold part of that tract of land to William Blewitt for 100 pounds. Later Thomas Dockery Sr. owned the tract where Cartledge Creek Baptist Church now stands.

There is a Cartledge Creek in Anson County, North Carolina

Anson County, North Carolina

Cartledge Creek

So perhaps in was in the 1760 range that Edmund III migrated to Georgia. Edmund is listed as a Captain in the Second Company, Augusta Division of the Georgia Militia commissioned on August 20, 1762

References:
Colonial Soldiers of the South, 1732-1774
Carolina Cradle: Settlement of the Northwest Carolina Frontier
Settlers of Maryland, 1679-1783. Consolidated Edition
History of Anson County, North Carolina, 1750-1976
Formation of NC Counties 1663-1943
A History of Richmond County
South Carolina Marriages, 1641-1965
Some "Early" South Carolina Marriage Records: 1641-1799