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I disagree with lennybriscoe in explaining why you wanted to cheat- even though you may not mean it that way it would sound like an excuse. I think you just saying "there is no excuse" is good enough. That being said, I think your last paragraph is too wordy and abstract- I think keeping things to the point and saying that you've learned your lesson and have changed in the most concise way possible makes things more convincing.

OP could just say something similar to "I didn't prepare well for the quiz and I wrongfully brought a cheat sheet with me to the test." While you wouldn't want to come off like making excuses, I agree that this draft sounds formulaic and thus, a bit unconvincing.