Sheep and goats and...a sheep with a sheepskin cover?

[Merton]

We have indeed been made alive in Christ and without Him we are nothing and are capable of nothing good (cf. John 15:4). We are not by ourselves intrinsically good or righteous. We partake of His love and His righteousness, having been united with Him through faith.

I agree that we "partake" of Christ's righteousness.

Galatians 5:22-23
But the fruit of the Spirit is love, joy, peace, forbearance, kindness, goodness, faithfulness, gentleness and self-control. Against such things there is no law.

This fruit is Christ's own righteousness in us. The way a branch on the Vine produces good fruit.

The expression "law of Christ" in the NT is restricted to two Pauline passages (Gal. 6:2 and 1 Cor. 9:21). There is nothing in those passages indicating that Paul thinks of a law of commands and regulations similar to the Mosaic law. Most likely, it is an expression used to calm the senses of the legalist Galatians who accused Paul's gospel of antinomianism. Instead of being bound by a law with detailed regulations, we are led by the Spirit (cf. Rom. 7:6) to carry out good works (called "the fruit of the Spirit"). At best, Paul thinks of the general command to love one another that Christ gave His disciples. Compare with what John writes in his first epistle:

And this is his command: to believe in the name of his Son, Jesus Christ, and to love one another as he commanded us. (1 John 3:23)

I completely agree that we have no written law to refer to and to observe in the Mosaic fashion. We have been freed from the written Law. Instead, we are moved from within ourselves by impulses towards love, peacemaking, forbearance, kindness and goodness, for example (the fruits of the Spirit), and these impulses effectively tell us what to do, and so can be considered a law.

For example, we can have evil desires rise up in us. If we follow them, then they "reign" in our lives, and we "obey" them as the law to do these things. It is "the law of sin at work within me" (Romans 7:23). There's no written law, but desires of themselves are their own law telling us what to do.

Romans 6:12
Therefore do not let sin reign in your mortal body so that you obey its evil desires.

But when we follow the life that the Spirit is placing in us, then we let the Spirit reign and follow this life as our new law. If you see someone in need, and the Spirit fills you with compassion and gives you the impulse to help that person...then that is the "law of the Spirit," and Christ's life living in you, and the "law of love." The written command in 1 John 3:23 is simply pointing us towards what we will find within ourselves and telling us to follow it: "Find love within yourselves, which will come from Me through the Spirit, and let this love reign through your actions in your life."

Romans 8:8
Therefore, there is now no condemnation for those who are in Christ Jesus, because through Christ Jesus the law of the Spirit who gives life has set you free from the law of sin and death.

Romans 8:9
You, however, are not in the realm of the flesh but are in the realm of the Spirit, if indeed the Spirit of God lives in you.

The "realm of the Spirit" is the kingdom governed by the law of the Spirit.

Christians are not saved from sinful desires, in the sense that they do no longer have such desires. But we are called to suppress those desires.

We continue to have such desires. And they can also overcome us and sidetrack us from time to time, even often. But we are no longer captive under them. We have been rescued from them. The written Law was powerless to effect that liberation, but the Spirit has the power to set us free from them. Thus, we can now not allow them to reign, but can follow the Spirit instead.