All time you’ve got people like Bob Dylan or Leonard Cohen who are poets, basically. Say what you want about their music (Cohen’s is pretty ordinary IMHO) but their lyrical output is in a different league – very few singer songwriters can stand comparison.
Morrissey is in the pantheon, no doubt, but I don’t think of him as a very honest songwriter like the OP says. He’s more arch and clever – like a melancholy PG Wodehouse. Still one of the best of the last 30 years say.
Personal favourite is Will Oldham (Bonnie Prince Billy). Not listening to him as much these days but he’s the modern songwriter I’ve liked best.