The maiden name of Mullins is from the birth record of her son James, and the death certificate of her son Jesse. (The death certificates of her other children do not specify a mother’s maiden name.)

1880 census of Magoffin Co., Ky, Burning Fork precinct, enumerated 4 June, lists on p. 19 as family #154:
Adams, Samuel J., 56, farmer, b. Ky NC Tenn; Mahala, 48, wife, b. Ky Va Tenn; Farmer J., 24, son; William S., 21, son; Louisa, 18, dau; Malinda V., 14, dau; Henry P., 7, son. All children born Ky Ky Ky. Roark, Elizabeth, 75, mother, widowed, b. Tenn -- --; Williams, Frank, 25, servant, divorced, cooper, b. Ky Ky Ky.

Many family trees on Ancestry.com attach Samuel Johnson Adams, of Morgan and Magoffin counties, as the son of William “Uncle Billie” Adams (1802-1881), son of Stephen Adams and Mollie Webb, and pioneer settler of Magoffin county. But this seems unlikely, considering that (1) he does not appear on the list of William’s children in the biographical sketch in Perrin’s History of Kentucky, and (2) judging by the 1880 census, Samuel’s mother outlived both her husbands, since she is recorded with him as a widow Elizabeth Roark, age 75, in the 1880 census.

Note also that Elizabeth’s marriage to James Roark was in 1839, when she was about 33. So it appears most likely to me that she first married an Adams, and that he died rather young. The fact that both “Uncle Billy” Williams and Samuel Johnson Williams named sons William Smith Williams does suggest a tie to that branch, though. So I suspect that his father may have been a brother of “Uncle Billy” Williams probably too young to appear in the will or estate settlement of his own father. (Those estate records have been one of the main sources for reconstructing the early Adams/Caudill families.)