Those who knowingly reject the atoning Messiah (the priests of Noah) are "carnal and devilish." While we do not know if Abinadi has specifically accused the priests of succumbing to the temptations of the flesh, certainly Mormon characterized them that way (see Mosiah 11, especially verse 14). If there was no prior accusation on Abinadi's part, he is probably reciting a sentiment that was common among whatever part of the population did not embrace Noah's reforms.
As Abinadi builds to his climax, he reinforces the mission of Christ from yet another angle. Since the issue at hand is the requirement that even the law of Moses recognize this atoning Messiah, Abinadi begins with another point of agreement between himself and the priests, the Fall of Adam and Eve. As part of the text on the brass plates, this must be something that the priests believe and teach.
What Abinadi points out is not the Fall from the grace of God, but the Fall from innocence before God. His emphasis is state of sin into which Adam and all mankind fell. First, Abinadi emphasizes that Adam and the family of man now knew good from evil. This is essential, as Abinadi has already stressed the salvation of innocents (Mosiah 15:24-25). It is in a state of knowledge that Adam and mankind have fallen. Because they are under the sway of sin, they are also under the influence of Satan.