If you say I/He FELL asleep, then you mean that I/He closed their eyes and went to sleep.
You cannot "fell sleepy" and you cannot "FELT asleep". Those both make no sense and are grammatically incorrect.
FELL is the past tense of fall, whereas FELT is the past tense of feel. And so you cannot "fell sleepy" (sleepiness can only be feel by someone) but can feel sleepy as well you can fall/fell asleep but can't feel/felt asleep.
SLEEPY (adjective), means: needing sleep; ready to go to sleep [ syn. is Drowsy ] whereas ASLEEP (adjective), means: sleeping [ the opposite is Awake.]
[From: 8thnote, Tofail; WordReference.com (03 Jan 2017)]
[From: 8thnote, Tofail; WordReference.com (03 Jan 2017)]