I would slightly modify the Title staement in my own context, I need the rhythm first, not necessarily a tune, or melody, but I have to have a tempo and a sense of how many "feet" in each line. I am a Words person First and foremost, I have always written poetry, and songs are just poems with a tune defining the rhythm. As someone said above songs do sometimes "write themselves" especially if its a subject area that I have immersed myself in and gathered, over a long period, a large amount of information. I rarely get words and tune together. On one memorable occassion I was on my way to a Folk club on London Underground, when suddenly i got the first verse, chorus and Tune all came into my head together, so I dragged out my Notebook (most songwriters carry one all the time, some of my best work has come while I was walking or travelling) that was the words yakemn care of my real problem was the tune, I do not write music!!!. so I just kept singing it to myself over and over again, until I reached my destination, wher I went straight into the toilets and sang it into my recorder, which I luckily had with me!!!
I wish I could write melody, and I envy those that can, its why sometimes I collaborate with others, I do the words and they do the tune,as in "Thiepval" and "Channels" both with Linda Kelly, visit Hissyfits website and you can judge for yourself whether we suceeded or not.