Do you start with the tune, then add harmony, or do you develop the melody from the harmonic progression?
From an educators point of view, I say both. Melody and harmony are interwoven, each one motivating the other. Play you melodies AS harmonic progressions, because they are. Then lighten them up with the appropriate chord support.
I suggest you take a piano class because it's much easier to write/arrange on the piano. You can see the melody and harmony, and if you study piano a while, you learn to voice that harmony with the melody, thus connecting them in a linear way and move forward.