Were the Corinthians believers or unbelievers when Paul wrote to them?

They were believers, not unbelievers, though carnal and fleshly. Paul addresses them as believers who were not growing as he expected. Their carnality appeared through divisions, envying, strife, and loyalty to pet people within the congregation, distinguishing them from the unbelieving segment.

Les explains that God sees the human race in three segments: spiritual believers, carnal believers, and unbelievers. The Corinthian believers for the most part were believers, but they were carnal and still fleshly. They were not growing as Paul expected them to. Their carnality showed itself in divisions, envying, and strife, as they had pet people they wanted to follow. This occurred even though the congregation met in the house of Chloe and was small enough to meet in a home.

1 Corinthians 3:1

And I, brethren, could not speak unto you as unto spiritual, but as unto carnal, even as unto babes in Christ.

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Were the Corinthians saved or unsaved if Paul calls them carnal?

The Corinthians addressed as carnal were still saved believers, not unsaved. Scripture distinguishes spiritual believers, carnal believers, and unbelievers. Paul calls the Corinthian believers carnal because they remained fleshly and were not growing as expected, yet they were still believers, though immature and divided.