Do Christians still have the old flesh even after receiving a new nature?
Yes. When old Adam dies, God imparts a new divine nature that cannot sin because it is divine. Yet believers still have the old flesh and the old Adam, which remains capable of tripping them up. So the new man cannot sin, but the old flesh persists.
Les explains from verse 17 that if any man be in Christ, he is a new creation; old things are passed away, and all things become new. When old Adam dies, God imparts a new divine nature that cannot sin because it is divine and placed there by an act of God. But the old flesh and old Adam remain, still capable of tripping us up. So the new man cannot sin. Les also connects this to verse 18, where all things are of God, who has reconciled, and God wants to bring the whole human race back into full fellowship.