Absolutely. I've made the same point on other threads. Technically there were no transgressors like Adam and Israel (from Moses forward). Yet we are all slaves to sin and death. This is not a "sin nature" but a human nature under the headship of Adam. Christ was made flesh, in the likeness of sinful flesh. He was tempted, yet without sin. We are tempted by what is in our nature. Jesus was tempted by what was in His nature (as a man). This is not sin. Sin is a matter of the will, not the nature.
There is no passage of Scripture that states Adam's nature changed before, after, or during the Fall. Rather, James tells us we are tempted by things of the flesh and when we give in to this temptation sin is birthed. As you so aptly put it, when we put our desires...ourselves....in the place of God or God's will this is rebellion and sin.