Going through the Carter material posted by Richie brought a few thoughts up from the mire of my brain.
A number of songs by the Carter Family are under copyright; while it is true that many are based on old parlor or folk songs, the text revisions, and often simplifications to the melody, amply qualify them for copyright.
Singers wishing to use the songs without observing copyright must go back to a pre-Carter version, or write their own verses to a melody that is not a Carter revision of the original. How successful they are in doing this may also depend on a publisher's opinion and/or legal adjudication.

The lyrics posted by Richie makes it evident that A. P. Carter was not just a song borrower, as some have maintained, but was competent as a composer and arranger, and had a sure feel for re-writing old material into a form that fit his time and feeling.

Also evident is that parlor music, song sheets, and the cheaply printed songsters, reached into rural areas. Those who could afford the printed music obviously shared and traded material. Some songs from the mid-19th c. were handed down in folk form, but I doubt that this would account for the majority of the old parlor songs played by the Carters and other rural musicians.