Singing a capella is the main way I make music.
Just recently I have learned the value of singing with a more strict time: it drives the song. It also makes it easier for others to sing with me.
One can sing "in time" without turning it into a dance step(American Bandstandization) by keeping the sense of time internally, and when you slow down or speed up while "telling the story" keep the internal metronome going. This is called "rubato". If one keeps the beat going through the rests, and the verse to chorus breaks, the song builds much better than if you pause while you recollect the next stanza. Trust me, I know.
I think the most important part of singing the song is to tell it fresh from your own heart/self/experience each time you do it.
Blessings,
Barbara