I've been reading a lot of fiction derived from Appalachian folklore lately - specifically, the works of Manly Wade Wellman and his tales of Silver John, and also the Hillbilly comic (please forgive me for linking to Wikipedia and tvtropes, respectively) - and I am quite frankly fascinated by them. Obviously, both these sources are fictionalised, so there's a fair amount of author-intended bias in the work, but underneath that I can see clear Old World influences in the work (and Wellman is outright explicit about how European culture "invaded" North America). In Hillbilly, there are witches that wouldn't be out of place in Macbeth over every hill; animals talk, magic is real, and silver burns any evil creature to the touch.
So what makes Appalachian folklore distinct - not just from the Old World, but from surrounding regions?
Manly Wade Wellman is from South Carolina, and during the 1940sand 50s made a bit of a splash at Weird Tales and The Magazine of Fantasy and Science Fiction for his series characters of occult detectives, Judge Keith Hillary Pursuivant, John Thunstone, Lee Cobbett, and John - sometimes known as "Silver John" or "John the Balladeer." While all of Wellman's stories partake of his idiosyncratic style, it was the minstrel tales, set in the Appalachians, which have most caught the popular imagination, as they are a novel setting compared to the typical urbane occult detectives like William Hope Hodgson's Thomas Carnacki, Algernon Blackwood's John Silence, or Seabury Quinn's Jules de Grandin.
Works directly inspired by Wellman's character include David Drake's Old Nathan (1991), Mike Mignola and Richard Corben's Hellboy: The Crooked Man (2008), Eric Powell's Hillbilly (2016) - but here we should start to draw a distinction between genuine Appalachian folklore and what Wellman and those inspired by him are doing, which could technically be considered "fakelore," although the intent was to entertain, never to deceive someone into thinking it was real.
The Appalachians were a very uncommon setting at the time for weird fiction; the region was often seen as still remote, primitive, a relic of a past century, the inhabitants uneducated and often inbred, with distinct dialects and customs - and Wellman presented a more positive picture of these folk. Wellman's great innovation was in the setting and characters: he presented the mountain folk as rustic and often poor in terms of cash or material goods, yet predominantly honest, intelligent, good-hearted and kind people, with their own pride at their accomplishments and a great love of family. His heroes are open-minded to other religions, but invariably the titular characters are Christian; they survive - and sometimes triumph - over evils by faith and strength of character as much as charms or specific weapons (often, strength of arm helps too).
The folklore and occultism in the Silver John tales in particular is a combination of real-life occultism and stuff that Wellman made up. Most of it is not particular to any real-world Appalachian folk tradition, although Wellman incorporates lots of bits and pieces - he famously used real works from the Pennsylvania Dutch hexerei tradition, the Long Lost Friend of Johann Georg Hohman and the Petit Albert, in place of fictional grimoires like Lovecraft's Necronomicon for the most part, and he often borrowed bits and pieces from different occult traditions - snatches from the Bible, ancient history, contemporary paranormal studies, whatever seemed appropriate - which gives the body of his work a bit of an eclectic taste.
On top of that, Wellman completely fabricated some of the dangers that John and his contemporaries face. There weren't any druids in Georgia, as in the novel The Old Gods Waken (1979); "One Other" in the story of the same name is nearly a Lovecraftian horror; the varied critters from "The Desrick on Yandro" are original creations of Wellman's own, not connected to any Appalachian folktales. Other enemies (vampires, werewolves, etc.) are more traditional, and typically resolved by familiar means, sometimes with a twist.
There is a peculiar twist in the Shonokins, a race of inhuman magic-workers that pass themselves off as a Native American people. Wellman was inspired by the idea of the "Mound Builders," a precursor race that was thought by many Americans to had predated the ancestors of the Native Americans, eventually being defeated and driven out by them - this invasion-and-colonization narrative was used as justification for imperial conquest of North America and to cement the idea of Native Americans as "savages" incapable of feats of construction such as seen at Cahokia, Ocmulgee, Moundville, or Fort Ancient - but this view was being disproved by archaeologists in the 1920s and 30s. Wellman's fictional take was, to his surprise, buoyed by eager readers who claimed it was real, even though he had made it up.
Scholarly is the report of Stuart Morton Boland, San Francisco librarian and adventurer in many lands. His studies reveal that many Indian tribes, North and South America, knew and feared the Shonokins—“beings like men who walk the earth as men but leave wherever they walk a place accursed.”
So there is a small core of genuine folklore in many of Wellman's tales - it just isn't necessarily authentic Appalachian folklore, although books like The Long Lost Friend and The Sixth and Seventh Books of Moses had many printings and a wide distribution, being influential from New York City to New Orleans and all places in between. Most of the verisimilitude was devoted to the settings and characters, the latter of whom dressed and talked as good, honest, mountain folk did, at least in Wellman's estimation.
The works that followed borrowed more of the setting and feel than anything else specific; the idea of fantasy being used by these rustic characters in a very rural setting. Old Nathan for example has the titular character gain the ability to talk to animals and use magic after getting his testicles shot off in the Civil War, and has a silver pocket-knife; the use of silver as a weapon against evil was a common theme for Wellman, who gave Judge Pursuivant and John Thunstone silver sword-canes, and "Silver John" a guitar with silver strings. The Crooked Man employs a combination of real folklore and "fakelore." Hillbilly - which word Wellman would not have liked - is entirely original, the whole setting being much more fantastic than Wellman's creation and riffing off elements of Powell's other work with The Goon and Billy the Kid's Old Timey Oddities.
In all of these works, the key is the approach to the fantastic and the characterization of the people and places rather than any specific folklore elements. You could draw a comparison with Southern Gothic literature that adds supernatural elements to a rural setting, although in general Wellman's stories are more upbeat and action-packed - good does generally triumph over evil, old ghosts and feuds are laid to rest, etc.
So, that goes more into the underlying premise of your question than what constitutes the actual folklore of that region, maybe u/itsallfolklore might have some additional info on that.