Wow.I have never really considered "the" method of songwriting I use. Depending on the song,I believe there are several.

1) write the lyrics first. This might be the most unusual for me. When I write a poem, it is usually conceived and executed as a poem, and the rhythmic structure, rhyme sequences, and verse/chorus pattern is lacking.

2)write the melody first. This is more common. I sometimes noodle on the guitar and come up with a chord sequence that is intriguing. Then I come up with a chain of words that seem to fit the melody, for example "the dawn disturbs the darkness", and then extrapolate to write the rest of the tune and lyric.

3)Words and melody arrive as a piece. Riding my bicycle once, I found myself singing, over and over, "come down with me to the rendezvous/ you can sell your skins and kick off your shoes" to a specific tune. It was easy to then add the second couplet "plenty of grub, and plenty of booze /when we get to the rendezvous." The rhythm was and is obvious, and I put it in standard 1-4-5 chord structure. This became the chorus/refrain. The verse needed to be different, so I started each verse stanza on the 4, but kept the 1 and 5, and reverted to the same chorus melody to conclude each verse. The rest of the lyrics, the verses, had to be about a rendezvous (obviously), and so I did some online research and kicked up some place names and famous mountain men to give it an aroma of authenticity, and inserted these into the verses.

I don't know that anyone else writes songs this way, but using methods 2 and 3, I'll bet I could sit down and come up with a complete song in a half hour with only a moment's notice. Might not be a good song, but...