If you look in almost any of the major collections, you will find several variants on almost any of the songs. Many of them are fragments.

A classic example is Sir Patrick Spens. The version we were all presented with in school is woefully incomplete if you go to the collections of either the Appalachian or Anglo/Scottish collections. On the other hand, it is possible to create a concordance from these which recreates something which may be closer to the original--all xx verses of it.

Is the song frozen by the act of publication? No, but at least one version of it is preserved.