Good tips Melody. It needs a fair degree of ear training to do, listening not only to your vocal and it's pitch and tonality, but also listening to a generally louder vocal singing a melody that your own pitch is relative to.
Worth mentioning is that this is an exercise you have to do several times, as you are unlikely to hit the harmony line, if you are also making up the harmony line. It takes experimentation and then once you have decided upon the harmony line, that is when the practice begins. That is why picking something that already has a good harmony is useful. The hard work is done for you regarding writing the harmony line to be sung.