I disagree with some of the other answers. I have heard “drop off” used to mean “fall asleep.” But it is not very common.

"Drop off" isn't typically used for sleeping. "Drop off" is usually used to state that you are taking a person to a specific location and leaving them there. For example: Tomorrow, I will drop you off at the mall. For your second sentence, you would say "fall asleep" rather than "fall sleep."

'Falling asleep' is when you're about to go to sleep in bed or somewhere. 'Dropping off.' Is when you are taking something/someone to a certain location to leave something or someone.

Are you sure you don't mean "drift off?" This phrase is related to sleeping. "fall asleep" simply means to begin sleeping. "drift off" means that you sleep without meaning to; it's unintentional. For example: "I drifted off during class today, so my teacher scolded me." The person fell asleep without intending to during his class, so his teacher yelled at him. Does this make sense?