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I'm usually the other way round. I write most of the lyric first and then starting feeling around for the changes on guitar. For me its all intuitive based on how the words make me feel and what I hear in my head (and there are compromises).
I do occasionally work right side up, or music first. My guitar player is a machine and will send me stuff that is almost whole, like a demo with rhythm guitar, bass and drums, and I will throw it up on garageband, plug in a mic and just babble, recording it as the brother above me described. After 3 or 5 takes I can usually hear a framework for the melody lines and -- because I write all the time -- I generally have a quick sense for syllables and timing, so -- I begin to chip away, picking around the edges of the lyric until I get a line completely worked out. Once I've got the lyrical DNA molecule (or maybe one for each "part" of the song) I quickly fill in the remaining lyrics, trimming and revealing, until the whole thing comes together. It doesn't always lead somewhere interesting; part of songwriting the lyrics is inspiration, but the rest is craft. I try to stay as prepared as possible. I keep a stack of handwritten stories, never-used lyrics and poems by my laptop when I'm writing lyrics, in case I need quick inspiration.
I read a ton. I pay attention to lyrics and tend to look at a song as a marriage between the different parts, including the lyrics, as well as the way they are delivered, in s song. I listen for sneers, smiles, eyeball rolls and other shit normal people probably aren't consciously aware of in a singers voice. I think about how an emotion will color or change a performance of a song and try to project that emotion -- like an actor -- when I am singing it. Again, there are compromises, but for me -- writing songs is almost a religious experience, and I treat the lyrics accordingly.