You really have the right idea with this. Usually you would start by making rhyme poetry, because it makes your poems flow better. And it makes it easier to read. However, poetry isn't limited to rhyming. Also, using synonyms is also a good idea. During the next line of every poem, you should start with a capital letter, even if it isn't a new sentence.
Your poem is actually very good, and I think you're a natural.
Rhyming is a good way to start when writing better poetry. For instance (with this part of your poem);
[original]
Their song of agony makes me cringe.
Even when the wind is still,
I can hear the echoes of their screams.
[with rhyming]
Their songs of agony make me shrink.
Even when the wind is still,
I can still hear the echoes of their shrieks.
So you can see how I kind of used rhyming to rewrite that stanza of the poem. I used synonyms so they rhymed.
You did a very good job, and i just wanted to offer some advice that maybe helped you.