For my last combo, which was active roughly between 2009 and 2018, I was the composer and arranger. I did everything old school, writing arrangements with pencil on paper.
I would typically start out with some some sort of overall mood/vibe I was going for, and then come up with a motif which I would expand. Sometimes it would branch off and result in two different tunes.
While I did write a few things in a more conventional song format, with 8-bar phrases, most of what I wrote involved thematic development, so there would be different sections beyond the typical AABA. Some things were completely arranged top to bottom, and some included stretches with improvisation.
I would typically compose at the piano, but I have a good ear, so I can also compose when I'm taking long walks and record myself on my cell phone.
Keep in mind I was working in a peformative genre, in which the goal was create music designed to set a certain mood.