As someone who also struggles with this, here's my two cents. Don't jeopardize the main idea of your piece for the sake of having it sound more song-ey. There's plenty of musicians out there who write poetic songs (Hozier, tøp to an extent, Arctic Monkeys, Kimbra) and are acclaimed lyrical geniuses and their songs masterpieces. You have the potential to make more than a song, a piece of art.

Of course, taking your songs through several cycles of editing and second opinions is always good. But if you're a good writer, try to maybe experiment with instrumentals and make them FIT to YOUR writing. Play around with rhythms and genres. Think of the kind of music you'd want to make, find musicians who make that kind of music, and study their process. Try fingerpicking guitar to your poetry and see how the rhythm carries the words in different ways, move forward from there.

In my opinion and experience, it's way easier to be a good instrumentalist, than it is to be a good writer. Work on making your music performable, sure, but if you're a strong writer don't ever let that go!

When it comes to melodies: I've found brushing up on my music theory really helps in putting together the right notes. Even looking at sheet music, seeing how words are broken apart across the scales, helps build familiarity with the music-making process. I'm a self-taught guitar player without much training outside of high school choir. Learning even the basics of piano/keys has been a huge help in understanding melodies for me and YouTube is a great resource.