No terms for sin are used in these chapters. However, what is to me some key verses on Paul and law in the life of a believer appear in 9:20-21:
To those under law [I became] as under law--although I am not myself under law--so that I might gain those under law. To those without law [I became] as one without law--although I am not without the law of God but am "enlawed" of Christ--so that I might gain those without law.
I hear in these words some of the background to the Corinthian slogan--"All things are permitted to me." But here (and in Romans) Paul makes it clear that not being under the law is not a moral free for all. God forbid! While he may not be judged by the Jewish law in its ethnic particulars, there is a core law of God, law of Christ under which he is still "enlawed." This interpretation of "under law" and "without law" bear further support from other texts, but for now I throw them out for the time being.
So here I would argue Paul believes that there is a law that applied to believers even though they are not "under law" (cf. Rom. 3:31). The remainder of 1 Corinthians 9 demonstrates that it is possible not to attain salvation even after being a believer. Paul talks about how he disciplines his body (note the connection to flesh language elsewhere in connection to sin) so that after preaching to others, he will not be adokimos, disqualified, unworthy.
If anyone has any doubts about what he has in mind, we need only read on into the next chapter, remembering that the chapter divisions are not original to 1 Corinthians. In chapter 10, Paul uses the example of the wilderness generation to show that it is possible to be in the people of God and in fact be overthrown in the desert (10:5). The sins of idolatry and porneia that Paul mentions subsequently are sins he has already said will keep a person from inheriting the kingdom of God. There can be little doubt, therefore, that he is suggesting no one--he or the Corinthians--are guaranteed the kingdom no matter what they do, that neither he nor the Corinthians are "eternally secure."
So Paul implies that sin can disqualify a believer from the kingdom of God, particularly the sins of idolatry and porneia here. He is not arguing from the standpoint of "one sin you're out," however, but his arguments target sustained sinful behavior that goes uncorrected. Such temptation can be avoided, for "no temptation has overcome you but what is of a human sort. But God is able, who will not allow you to be tempted more than you are able to bear, but will make a way of escape with the temptation, that you will be able to bear it" (10:13). This is a plural "you," but it clearly indicates that sins of the sort he mentions are avoidable, indeed that they must be avoided.