I don't know anything about the book so I'm commenting without knowing the intent of the author or content of the book.
can someone who truly converted begin to fall back into old sinful habits again, or is this the sign of a false conversion?
Yes, we can all fall back into sinful habits but that does not necessarily mean we are false converts. If you never fell into sin I would say you probably don't know what sin is.
Paul wrote, "I find then a law, that, when I would do good, evil is present with me." We are saved by the righteousness of Christ alone. Nothing else. We trust in Christ alone and not our performance of religious duties no matter how important they are. As you can see from Luke 17.10 even if we do ALL that is asked of us we are still unprofitable servants.
"So likewise ye, when ye shall have done all those things which are commanded you, say, We are unprofitable servants: we have done that which was our duty to do."
The London Baptist Confession of 1689 reads,
"And though they may, through the temptation of Satan and of the world, the prevalency of corruption remaining in them, and the neglect of means of their preservation, fall into grievous sins, and for a time continue therein, whereby they incur God's displeasure and grieve his Holy Spirit, come to have their graces and comforts impaired, have their hearts hardened, and their consciences wounded, hurt and scandalize others, and bring temporal judgments upon themselves,
yet shall they renew their repentance and be preserved through faith in Christ Jesus to the end.
(
Matthew 26:70, 72, 74; Isaiah 64:5, 9; Ephesians 4:30; Psalms 51:10, 12; Psalms 32:3, 4; 2 Samuel 12:14; Luke 22:32, 61, 62 )
If you stumble and are never renewed by the Holy Spirit to repent...well, now you have a problem.
If it is possible, how should we deal with those who have fallen back to some old ways?
It is important the Elders know what's going on with this individual. Prayer is important. Take them to scripture privately and correct them. Preach the Gospel to them. Trust in God.
Yours in the Lord,
jm