The problem of worldliness can be found in the parable of the sower where there is no way that these are four types of Christians. Everyone really starts out as the first three kinds of soil and any true Christian today had to pass through soil conditioning before they become the good soil hearer. The cares of the world prevents people from ever getting saved because they care too much about it. The thorny soil has people who care too much about the world. Sometimes, before a true convert becomes a true convert, Christian preachers must continue to soften the soil, get rid of the bedrock and remove the thorns that is preventing the Word of God from making manifest. Because Christians went all the way from unsaved hearers to becoming receptive hearers, it will eventually create a non-comformity to the world as a result of getting saved.
When one becomes a Christian, they are already out of the world and not in it. John 15:18-19 confirms the reality that when you are a Christian, you are not of the world and that the world hates those who are not of them. Romans 12:2 addresses Christians should not be conformed to the world because they are not of the world but are called to be called to be in God's perfect service. When a person gets saved, this desire for non-worldliness increases as a result of sanctification. There is no way that any true believer would feel comfortable when they try to fit in and they would feel better if they were rejected by the world. The Christian's calling is out of the world because this world is not their home.
Why do I doubt those worldly people who claim to be Christians to be Christians? When God saves a person, He does not merely '"save" that person but instead, He saves that person from their sins and not in their sins. When there is grace, there is power against sin. Titus 2:11-14 shows that the grace of God is transformative so there is really no way a Christian can become perpetually barren. If they were unteachable before, God's grace makes them teachable. The grace of God is greater than the sin of the believer because the believer's victory over sin is a result of Jesus' mission to save people from their sins (Matthew 1:21). I have no reason to believe that I can be saved and still live like a devil because that is just impossible. I may not be sinless but certainly by God's grace and not of myself, I am in the road towards perfection and I refuse to take credit for my new life because as Philippians 2:13 says, it is God that works in me and not me working in me.