From the 1770 "Plan of the Town & Port of Beaufort" by French surveyor and cartographer Claude Joseph Sauthier

Beaufort NC - Carrot Island, not "Cart Island"

INACCURATE: 
1) Folklore: Carrot Island was originally "Cart Island," due to pushing carts over to the island marsh, or leaving carts on the "island."
2) Wikipedia article on Shackelford Banks: "When John Shackleford first acquired his tract at Shackleford Banks, the island was known as "Cart Island," most likely after Carteret County." 

1733 Moseley Map
ACCURATE: Carrot Island was first noted as "Carrot I." on Moseley's 1733 map. 

Carrot Island was included in the 1734 will of John Shackelford: "I bequest my estate to my son James and his heirs forever also Island called Carrot." (By the way, the family name was spelled Shackelford, not Shackleford.)

In March 1790, Benjamin Stanton purchased Carrot Island from Nehemiah Harris; Stanton had a "fishery" on an elevated portion of the island.

Carrot Island was noted on many early maps, including 1844, 1850, 1857, and 1888. The 1854 map labeled the channel that flowed from the downtown waterfront to south of Carrot Island as "Carrot Island Channel." Before dredging in the early 20th century, Carrot Island was essentially tidal marsh with some elevated hammock land.
 
1844
1850
1857
1888
1854 noted Carrot Island Channel