If falling for your body's desire is a sin, how is it then that masturbation isn't directly a sin, if it is a desire of the body?
The flaw is in your premise. Sin is breaking a law of God. "Whoever commits sin also commits lawlessness, and sin is lawlessness" (I John 3:4). Satisfying the desire of the body is not necessarily a sin. If you are hungry it is not a sin to eat. "The righteous eats to the satisfying of his soul, But the stomach of the wicked shall be in want" (Proverbs 13:25). If you are thirsty, getting something to drink is not a sin. "Then he became very thirsty; so he cried out to the LORD and said, "You have given this great deliverance by the hand of Your servant; and now shall I die of thirst and fall into the hand of the uncircumcised?" So God split the hollow place that is in Lehi, and water came out, and he drank; and his spirit returned, and he revived. Therefore he called its name En Hakkore, which is in Lehi to this day" (Judges 15:18-19). Did God encourage Samson to sin by giving him what his body desired?
If you start with a false premise, you can make no inference about the conclusion. Once the premise is shown to be false, the entire argument becomes false.