President John Adams had no daughter Jane, and his family had no ties to Rowan Co., N.C. See BURKE'S PRESIDENTIAL FAMILIES, 2nd ed. (1981), and its much more recent update whose data I don't have to hand, for descendants of Pres. John Adams (and of his siblings).
More than likely Jane (Adams) Tennyson had a relative named John Adams whom some later person thought must be the same as the president because hey, the name was the same. Or perhaps she struck that later person as an impressive personality in some way. Perhaps someone whose judgement in other areas she had reason to trust had told her so when she was a young woman, and she told it in turn. Much else of what the later teller reported may have checked out because it was closer to his or her own experience, so people working generations afterward accepted the whole story without question.
Because of the famous presidents of that name, Adams families nationwide have been especially prone to "grafts." Kenn Stryker-Rodda, ed., GENEALOGICAL RESEARCH: METHODS AND SOURCES (often called "GERMS"), rev. ed. (2 vols., 1983), contains a chapter on "Tradition and Family History" in which the great genealogist Donald Lines Jacobus (1884-1970) gives a humorous but all-too-true account of how any Adams family, anywhere, can become related to the presidents in 25 years or less (1:13-14).
New England, not to mention the faraway American South, is full of Adams families with no connection to the "presidential" Adamses, whose progenitor Henry Adams came from Barton St. David, Somerset, England, to Braintree, Mass., and died there in 1646. In the 17th and 18th centuries, there was little genealogical exchange between New England (whose excess youth tended to go north and west) and regions so far south and out of the way as the inland Carolinas. For a Henry Adams bibliography, the 1898 work by A.N. Adams, as well as two important 1927 (J. Gardner Bartlett, on his English origins) and 1974 books are referenced in an ancient post of mine to this forum called "Ap Adams is a forgery -- throw it out."
Hope this helps. I am sure Jane (Adams) Tennyson had no need of grand "relatives" to make her a special person to her family and descendants.