Am I being too analytical or do other songwriters get hung up on this too?
No you're not, and yes they do.
I scrutinise every single line of my lyrics to make sure they not only scan well, but fit the content of the song, have some meaning and sound good when I sing them.
As far as I'm concerned this is what all the best songwriters do. Listen to Paul Simon. You'll have to listen for a long time before you hear a bad lyric.
But I think you need to close the distance between your melody and your lyrics to a certain extent. When I'm writing a song, I always have the melody first. But once I find some lyrics that sound and feel good, the melody almost always changes shape in a small way to accommodate them. These changes are usually subtle but actually have quite a lot of impact on the overall sound and feel of the song.
It stands to reason, if all the lyrics are well-phrased with no melodic conflict, your song is going to have more going for it.
But Boydie also offers good advice. Writing as though you're speaking works really well for songs. And with English being a naturally stressed language, it offers a lot of opportunity for you to find melody and rhythm in the words themselves, outside of the context of your song.
Which I know, leads you back to the original question. But I say keep working at it. If you want to hear how the masters do it, pull up the Paul Simon solo catalogue, Joni Mitchell, Tom Waits, Neil Finn from Crowded House (he's a bit more abstract, but very good match between melody and lyric).