I can think of several examples of people who usually sing unaccompanied taking a song which is perhaps best known in its accompanied version (perhaps the way the songwriter him/herself did it?): it is quite noticeable, that if the lines in the accompanied version finish with long notes, the unaccompanied singer will end to shorten these, which actually throws the rhythm of the song out: whether this is because they cannot hold the long notes or are afraid of "dead air space", I don't know, or could be deliberately just "making the song your own".
On the other hand, some of the best tips I have heard for unaccompanied singers are (from Gordeanna McCulloch) "read the words first, and then sing them as you would say them, to make them make sense" - and for songwriters, from Hamish Henderson - "let the words sit comfortably on the tune". And add to that avoiding having conjunctions and prepositions (and, of, the, etc) as long notes: keep the longer notes for the important words that need to be emphasised, and pronounced as they would be in speech.
As for songs with strong rhythmic or melodic "fills" between lines or verses, just sing/scat them, don't leave them out!