As for The Carter Family, the "Program for the Sixtieth Annual Festival of the Worcester County Musical Association," published in 1917, seems to provide the answer in an essay starting on page 70 of "The Fourth Concert" section.
"I stood on the River of Jordan" started out as a so-called "Negro spiritual," part of a genre of gospel music associated with (typically enslaved) African-Americans. At that time, "the colored people" (as the essay states) would pronounce "Jordan" as "Jerdon." In the African-American English of the day, this resulted in "I stood on the river of Jordan" being pronounced as "I stood on de ribber ob Jerdon."
While The Carter Family sings most of the song in a traditional white "old Southern" accent, they appear to have deliberately preserved the original African-American pronunciation of "Jordan." This may be because the song wouldn't have been recognizable otherwise. But it's also likely relevant that this wasn't really about the river in the Middle East. As Wikipedia notes:
Before and during the Civil War, the Ohio River was called the "River Jordan" by slaves crossing it to escape to freedom in the North via the Underground Railroad. More escaping slaves, estimated in the thousands, made their perilous journey north to freedom across the Ohio River than anywhere else across the north-south frontier.
As for the Doc Watson song, this blog post by James Anderson states that the original song ("I Am a Poor, Wayfaring Stranger") may have had a similar origin:
If the song arose among enslaved Black people, the text takes on an additional layer of meaning. As scholars have noted, spirituals often contained coded language. “Heaven” or “the promised land” referred to the place of rest for the Christian, but also to the freedom of the North. The “Jordan River” might be an allusion to the Israelites’ journey in the Old Testament (notably, as they were leaving their own slavery), but also could refer to the Ohio River, which separated slave states from free ones. References to trains (as in “This Train Is Bound for Glory”) provided a hidden reference to the Underground Railroad.
The pronunciation of "Jordan" as "Jerdon" is thus likely of a similar origin: again the song started out in African-American communities where the "Jerdon" pronunciation was common, particularly when used to refer to the Ohio River, and that pronunciation was preserved by later white singers.
This may not be the whole story; as in the case of Hamilton Jordan cited in one of the comments above, the "Jerdon" pronunciation may also have been established independently among white speakers. But the above suffices to explain its occurrence in the two songs cited.