Mainstream theological scholarship has in recent years been moving away from orthodoxy on the atonement, in particular away from forensic or legal views of the atonement. The idea that God is wrathful and holds out a just punishment over all those whose sins aren't atoned for has become the whipping boy of many a theologian. The preference now is to emphasize peace with God, new life in Christ, or the example of humility and service of Jesus on the cross, as long as there's none of this sense of legal guilt and responsibility for sin that Jesus somehow takes care of on the cross.
Evangelicals have tended to react to this by ignoring some of the other aspects of the atonement. Peace with God and new life in Christ are generally thought of as results of being a Christian, but it's not the first thing emphasized by many evangelicals regarding the purpose of the cross. Jesus taking the penalty for our sins is usually the primary message, and the polemic against those who deny this usually leaves out the real variety in the scriptures' language of the atonement. What did Jesus die for? I wouldn't deny that it's to satisfy God's justice. I wouldn't deny that it's to bear a penalty that I owe. It's also buying me back as a ransom, which isn't the same image as paying someone's debt, and that in turn isn't the same image as taking someone's death penalty. But not one of these images fully captures the sense in the scriptures that the cross was to give us new life, overflowing life, of a quality not found outside Christ. Not one of them fully captures the shalom or peace that comes from my being in Christ and from Christ being in me. This isn't just gettinng along, as 'peace' in English suggests. It's wholeness and health. I'm whole and healthy in Christ. It's already true in one sense and is being worked out over time in another sense, and the New Testament authors speak in both ways. But there's one element of all this that I think deserves special attention.
One sinner's prayer is "Deliver me from all my transgressions." [Psalm 39:8, ESV] I think too often evangelicals take this to mean just deliverance from the penalty for transgressions or sins, but I don't think that's what this prayer is asking for (or at least if it is it's not all it's asking for). In the context of the whole psalm, there are several things to ask deliverance from. One is enemies, and another is physical infirmity or illness of some sort. But the third is sin. This is a prayer of deliverance from the sins themselves. It's also plural, which means it isn't deliverance from sin as a principle of fallen human nature, which the atonement is about. This is deliverance from the actual sins one would commit. The true fulfillment of such a prayer is in the cross. There's one word that's used regularly in the gospels for both healing and salvation, and I think there may be occasions when it might mean both, because the concepts are tied together. There is healing in the atonement. Some take this too far and assume that the atonement means I can just assume God will heal me right now of whatever I feel like. Scripture speaks plainly against that idea (see II Corinthians 12l for instance). But there is healing in this saving act. Ultimately we'll all be healed of whatever is unhealthy or unwhole, and I have plenty of deficiencies that I long to be healed of.
But there's one thing followers of Jesus are promised healing from in the end (and in one sense are healed of now but not in the full sense). That's healing from our sins themselves. In the New Jerusalem, no one unclean, who does anything detestable or false, will be there (Rev 21:27). While I think this statement does intend to say that some who remain unrighteous will not be there, it doesn't mean that no one can be unrighteous but then be there. Otherwise only God and the faithful angels could enter. It means that those who do end up there will have been made righteous before they can participate in the resurrection community of the people of God. Those who will be there will have been saved from their sin in the sense of being healed of their sin (in addition to being no longer liable to pay a debt of legal penalty for it and any other aspect of the atonement you wish to add in). I think that's probably what I appreciate most about the cross that Christians today remember. It's not just being in a community of people. It's no longer struggling with motivations and preferences that at their very root are evil. Hence the longing at the very end of Revelation: "Come, Lord Jesus!" [Rev 22:20, ESV]