KimC, I think your intuition is right. My understanding (not legal advice, just private opinion) is that if the artist sings all truly PD then neither the syndicates nor anyone else has a claim. That doesn't mean no one will try to make a claim that the performer performed a copyrighted "adaptation" or "arrangement". And it's possible for copyrighted lyrics or melodies to be mistaken for public domain.
If the performer does his own music then there might, in theory, still be trouble from copyright cops. The author of the work doesn't always control the copyright. The following is a publication example, not a performance example, but for what its worth: John Fogerty was sued for infringing the copyright in a song he himself wrote. He successfully defended himself, however.