Brian, I'm not aware of any other text - of either "Bryan" or "Tom" - that includes the line about straw.

Nor am I sure that it's necessarily archaic.

"Tom" and "Brian" are so similar in structure that it was easy for stanzas to migrate from one to the other, but the available evidence is that it was usually in the direction of "Brian."

Probably, in part, because of the various broadside printings, "Brian" is frequently quoted and seems to have been the more common of the two since well back into the 19th century.