Ontologically man is a free moral agent; this did not change after the fall of Adam. Man's free will is as much a part of his constitution as his ability to reason. Jesus said the one who sins is a slave to sin and Paul told us to present ourselves as slaves to righteousness.
Jesus did not say that men are born slaves to sin, but that men become sinners after repeated sin - didn't your mothers tell you that as you lie it gets easier and easier? If you continue to sin you will ultimately become a slave to it and your nature or "way of life" will be one of sinfulness.
Paul's verse has a 'logical implication' that we are morally neutral and it is up to us as to who we will become enslaved to. Not many Christians ever practice righteousness; the old saying "it takes 21 days to form a habit". How many Christians are willing to relinquish a life of sin and become slaves of righteousness? How many Christians will even give this a half-baked effort? I would say very few will because of this horrible decree of original sin.
A sin nature is completely unbiblical and brought in to the church most likely by Augustine's Manichean deterministic philosophy.
Where Paul says in Ephesians 2:3 that so and so were Children of Wrath by 'Nature' does not carry an implication of some innate proclivity to sin. Paul elsewhere says to the Jews that 'we are Jews by nature and not sinners from Among the Gentiles'; so as we can see the word 'Nature' does not mean some innate proclivity that one is born with.
Finally, if God were to judge a sinner for sinning when He created that very sinner with a sin nature would be the largest act of injustice! This would violate the concept of justice that the bible teaches us and would violate God's character and His law.