It's very likely the existing broadsides had a heavy influence on oral versions, but to determine any likely precedents a full study would be needed. Dixon's sources were anything that came to hand and he rarely attributed his sources properly, though those he gleaned in his native Yorkshire Dales are usually the exception. Many of them are undoubtedly other print sources like broadsides.

An interesting aside. One of the Pitts copies I have has for its cut what looks like a portrait of Burns!

There is a copy of what looks like Burns' version in an Edinburgh songster dated 1806