Your poem is very cute and very ambitious. I think we can all relate to moments like these, when all of nature seems to be feeling the same way we do, and we start to see our own emotions around us in the behavior and events surrounding us.
I really like your subject matter.
I know you asked to critique the drawing as well, but I don't know that much about visual art; I'm an English/Creative Writing major, so that's what I'm going to critique. I hope that's alright.
Since your poem is so short, I'll go line by line.
"The crickets softly chirping after freshly fallen rain,"
This line, when taken in to context with the one after it, reads like this: "The crickets softly chirping after freshly fallen rain,/ I sat there and listened to the sad song they sang." The first line is written in present tense while the second is in past tense. This makes the poem read a bit awkwardly as the reader has to pause mid sentence to change tenses. To fix this, I would suggest changing the first line to read:
"The crickets softly chirped after the rain fell,"
But, we do want to keep the image that the rain has freshly fallen, since that is essential to the poem's message, so here is an example of something both grammatically correct, and it contains the images you are going for:
"The scent of falling rain lingered after the pour and mingled with the gentle chirping of crickets;"
This way, we have the images you created of the fresh scent of rain which lingers, and we have the image of the pouring rain, which will coat the image of the crickets' chirping with that slight air of sadness and longing you're going for, but in my example, it is grammatically correct, and has a few added benefits:
In my example (though you don't need to use my example, obviously, ), I've used assonance (the way that the same "ing" sound repeats in "lingered" and "mingled") to link the two verbs together, and thus to the same subject--this makes it subconsciously easier to associate two predicates with the same subject (plus assonance sounds neat when you read it aloud ).
I added a semicolon at the end of the line instead of a comma, as this is a complete idea in and of itself, but it will be linked to the next line. . .
"I sat there and listened to the sad song they sang."
Hmmm. Now, poetry is all about saying the most with as few words as possible, so accuracy and vivid, descriptive words are important. I would suggest either shortening this line or rewording this line and using it as an opportunity to add in more description. How is the speaker sitting? Are they lounging? Are they staring off in to space? Drooping their head down? Leaning on one arm? the smallest human gestures can give away tons of emotions--remember, an earthquake is just a shudder compared to the turmoil churning under the earth's crust.
I would also suggest to describe their song more accurately than simply "sad." Perhaps you could call it a "melancholy melody."
"Softly I hummed my own sweet melody,"
Hmmmm. Now, "softly" is not a particularly good word. I shall explain why. "Softly" means different things to different people, this means it is abstract. Also, (especially since your poem is so short,) it is not wise to repeat the same word twice so closely together, and you have "softly" in your first line.
I would suggest replacing "softly" with another insight in to the speaker's mind. Tell us what she is thinking of as she hums. I would also suggest not saying "sweet," as that is another abstract word that means different things to different people. Instead, think about how her song would actually sound, and try to describe it as accurately and concisely as you can.
"Waiting for the day that you would come for me."
This is the part that makes everything else in the poem relevant. You've done a good job of setting this up already. The ending of the poem is probably the most important part, since it's the feeling the reader is left with when they finish the piece.
In this case, I would suggest to be less obvious with the ending of your poem. Try to say this without really saying it--let the reader infer it from the images you present.
All in all, your poem is off to a great start! i enjoyed reading it, and I hope you find this critique helpful! I'm sorry if I was a bit long winded for such a short poem, but your poem is cute, and I think i will it.