Sumary: When writing vocal melodies, you'll always have to account for the singer's range, so don't worry about it. When building harmony/chord parts, it's a good idea to pay attention to the whole spectrum low-to-high. There's nothing wrong with backing instruments incorporating the same notes as the vocalist is singing.
If I understand you correctly, your current layers in terms of pitch, high to low are:
- chords (harmony) - highest
- vocal melody
- bassline - lowest
with an octave interval between the parts, so - for example - the root note of a chord is two octaves above the root played by the bass. Am I reading it right?
On the face of it, there's nothing wrong with the basic approach. The bassline - as the name suggests - should be the lowest part and with vocals, you've got to work within the singer's range (I presume that would be you).
As for the non-bass harmony (chords), do bear in mind that you have a lot of leeway in where you place your harmonizing notes and how you articulate them. A lot of the time you don't have to maintain strict pitch separation between the vocals and other instruments, because the human voice is something our hearing is especially sensitive to.
When writing the various parts that will make up the finished song, you'll want to pay attention to the entire spectrum of tones, low-to-high, in order to create a balance that is both pleasing to the ear - without obvious dead spots, such as having only very low and very high pitches with little in between - and appropriate to the genre you're working in.
My advice - as an experiment, if nothing else - would be to see whether you can move the notes that make up your chords around - drop them into the same octave as the vocals, for example; how does that sound? Or, perhaps, keep them low on the verses and high on the chorus. If you have several instruments/synth sounds/tones, see whether you can split the harmony up so you've got something going on in the higher frequency range (above the vocals) and something lower down, between the bass and the singer. Compare with similar songs that you like: how does the instrumentation contribute to the overall sound?
The only problem I see with separating parts by octaves is that you might end up with empty patches where it seems like "something's missing" (for example, the midrange might be dropping out between vocal phrases). In that case, you might wish to have your chords lower - with some higher-register additions as well - and maybe carve out spaces for the vocals as necessary. Too many things going on behind the singer will distract the listener, so it's probably best to exercise moderation with the background stuff when the vocals come in.
Finally, if what you have right now is exactly the sound you're looking for - go with it and don't worry. Whatever anyone (including myself) has to say about songwriting is "more what you'd call guidelines than actual rules".