Yes, by all means! Most contributors do sing to the original PD melody! That is the usual way.
However, there are a couple of circumstances where folks may choose to make up their own tune.
(1) In some cases, the words to an old carol appear in a carol book without any melody provided, perhaps on the assumption that everyone back then already knew the melody. In such a case, our LV contributor may create a suitable melody to fill this lack, as long as the original PD words are being used. For example, if you look back at previous years' carol collections, you may notice a number of older French carols to which Ezwa has supplied melodies, as the sources she used for the words did not include any melody.
(2) In some cases, the contributor has an idea of what sort of melody would fit the words, and the traditional melody or melodies don't quite satisfy. So the singer makes their own tune. Again, as long as the words are PD, and as long as the singer is not using a modern copyrighted melody, this is fine. Consider it a part of the "folk process".
But in most cases, contributors simply use the already existing PD music.
