"and sith he weeps" isn't changing "and see, he weeps" to make it sound more archaic -- it's a completely different thought or construction.The one says "his wounds are sick and for that reason he weeps." ("Sith" meaning "since" in the logical sense); the other says "His wounds are sick! Don't believe me? Look! He weeps!"
Given the way folk music tends to degenerate over time, "sith" seems more likely to be the original. When the word "sith" passed out of common parlance, an easier word was substituted.
Alex