If we confess our sins, he is faithful and just to forgive us our sins and to cleanse us from all unrighteousness. ~ 1 John 1:9
This verse is for Christians. The entire context of 1 John 1 demands it. A simple cursory reading of the chapter (all the ‘we’ references) confirms it. It was not written to the Gnostics even though Gnosticism may have been an issue John was combating.
Christians SHOULD be sin-conscious, because the work of the Spirit is that of conviction. People may mistake conviction with condemnation. The Spirit convicts, but does not condemn. Therefore we should be sin-conscious NOT sin-obsessed. And rather than contrasting sin-conscious and Jesus-conscious, I think a better way would be to say, we are sin-conscious because we are Jesus-conscious.
In no way does repentance of sin minimise the work of the Cross, nor the grace of God. In fact, it magnifies it. Because in the practice of repentance, we are reminded afresh of God’s grace that says, ‘where sin abounds, grace abounds all the more’. We appropriate the cleansing of His blood, we apply the Spirit’s empowerment in sanctification.
More than that, a proper understanding of 1 John 1:9 leads us not to fear (that our sins are unforgiven if we do not repent of particular sins) but to assurance (that God delights to forgive and will forgive anytime we respond to His Word and His Spirit in confession and repentance). This is our God! The God who delights to forgive.
A helpful article on repentance here.