How many semitones should I change to change melody but same chord

I guess you want to adjust the song to your voice range, while preserving the original key. So what you really want to do is compose a new melody for the song. There are several approaches, depending on what elements of the original melody you want to preserve

• You propose to shift the melody by a fixed number of semitones. This won't work, unless you shift by 12 semitones (an octave), or you're up for a modern experimental sonic effect
• You can use the original melody and only change the notes that are too low or too high. You can hear it often done by singers when they can't hit some top notes. Even Freddie Mercury did it, so there's no shame.
• You can compose a new melody trying to preserve the melody shape (i.e. when it goes up and down). The basic guidelines are:
  • use the same scale. Say, if the original song was in G major scale, keep using notes of the same scale
  • a good starting point is to shift the melody by a given interval, i.e. third (two scale steps) or sixth (5 scale steps). But there still might be issues because...
  • a good rule of thumb is for the melody notes on strong beats to be the chord tones. For example, we have a chord G (g-b-d) and the melody is b c d, ending on a strong beat. We want to transpose it by a third up, so we get d e f#. Now depending on the style, the note f# may or may not fit well with the G chord. Instead you may want to try a melody d e g. This way you preserve the original upward motion, and have a melody that agrees well with the chords.
• in case the words I put in italics don't ring a bell... I'd say you need to learn a bit more about music theory, in particular about harmony.
• an alternative approach is to play the original chords, follow your intuition and try to sing a new melody. Trust your ears. Record it. If it sounds good, it's good. If it doesn't, try different notes.

This a rather complicated question with a broad scope. The question is really how to compose a melody that fits against a given one. It's not as easy as (for example) singing or playing a fixed interval from the given melody. That could be done, but it tends to sound the group is singing with a thick texture rather than like an interesting embellishment on the original melody. One could sing a third (major or minor) or sixth (major or minor) or fourth or fifth above or below a given melody but that doesn't really add much (it's not wrong but can become boring).

A more interesting (but still achievable in real time) would be to sing a mixture of major and minor thirds and major and minor sixths below (or above) the given melody along with a few fifths and octaves. The reason for major or minor is to keep in the same key as the original. (If the original melody changes key, your added melody may anticipate the change but that's another problem.) Usually one limits use of the same interval to no more than 3 or 4 notes in a row; otherwise, the added melody loses independence and isn't really heard as something different.

It really doesn't work like that. You can change the key by a number of semitones, and sing the same intervals successively, but that's not what you are asking.

You seem to be asking about harmony. Which isn't another voice so many semitones away, singing parallel. The intervals change constantly. Do that and you will sound out of tune with people singing in the original melody.