The Bible speaks of the first Adam (as in Adam and Eve) and the last Adam -- meaning Jesus Christ. This would be important doctrine to the Puritans who were Calvinists and concerned primarily with the justification by faith. St. Paul writes about this in 1 Corinthians 15 and Romans 5.
Romans 5:17 says, "For if, by the trespass of the one man, death reigned through that one man, how much more will those who receive God’s abundant provision of grace and of the gift of righteousness reign in life through the one man, Jesus Christ!"
Reformation theologians such as Luther and Calvin, echoed St. Augustine and St. Paul in their reliance on this principle of justification. Adam failed in the Covenant of Works by disobeying God, but Christ came (God in the flesh) to fulfill the demands of the covenant. So Christ, as a man, lived a sinless life (the second Adam), and died on the cross (as a perfect sacrifice, fulfilling the demand of death place on the first Adam in the Covenant of Works.
So our justification is by faith, but faith in the work of Jesus.
I'm not sure about the Puritan reference you suggest about "The New Adam." There is no sense among the Puritan community that a preson is any longer under the Covenant of Works as the first Adam was. But Christians, justified by faith, are new creations, raised from death to life, quickened by the Holy Spirit (Ephesians 2:1-5).
I hope this helps point you in the right direction. Just remember, Jesus is the second Adam, completely human and able to sin as a human does, so his sinless life was an utterly fantastic accomplishment, one that He could accomplish because He, like the first Adam, was born without originial sin. The rest of us don't have that advantage. We are born dead on arrival (DOA), as the Epheisians 2 passages makes clear:
1 As for you, you were dead in your transgressions and sins, 2 in which you used to live when you followed the ways of this world and of the ruler of the kingdom of the air, the spirit who is now at work in those who are disobedient. 3 All of us also lived among them at one time, gratifying the cravings of our flesh[a] and following its desires and thoughts. Like the rest, we were by nature deserving of wrath. 4 But because of his great love for us, God, who is rich in mercy, 5 made us alive with Christ even when we were dead in transgressions—it is by grace you have been saved.