When I study lead sheet on the piano. The left hand plays the chord and the right hand plays the melody in the next octave up. Should this be the case when singing and playing too? So if I have a lower bass-baritone voice should I voice the chords differently to allow the notes of the melody to have their own octave?
This is something you have to experiment with and decide for yourself, on a case by case basis, and based on what sounds good to you, rather than based on some theoretical principles.
The principle of using different portions of the spectrum for different instruments tends to become more important as the number of instruments grows. In other words, if you are dealing with a lot of different instruments, like in the case of a big band or an orchestra, then placing them in different notes ranges is an important aspect of arrangement and orchestration.
Alternatively, if you have two or more identical instruments, e.g. two or three guitars, you also probably want to avoid playing in the same range.
But your case is the opposite extreme, it's only piano and voice, i.e. only two instruments, and with very different timbres too. Therefore I wouldn't worry too much about overlapping notes. I would choose whatever chord voicings sound the best, overlapping or not. I'd also record the whole thing, to listen objectively and evaluate what works well and what may need change.
"The left hand plays the chord and the right hand plays the melody in the next octave up."
That's one way to interpret a lead sheet. Or you can play a strong bass line, cover chords and melody in the RH as best you can. Or, if there's a singer, play bass and chords, let the singer take care of the melody.
Too little doubling of the melody can leave the singer floundering, too much can be restrictive. If this is a jazz performance, perhaps play no melody at all, or improvise a counter-melody, or play all the melody for the singer to scat around or...
Look at the repertoire of accompanied song - Mozart, Schubert, Britten etc. etc. for the many ways in which piano can support and complement a singer.