Simply to say, there is law and there is gospel in both the Old and New Covenants. In the Old, "Do this and live" (law) stood as God's righteous demand upon Israel and all men for perfect holiness, pointing to eternal life through perfect obedience. And it was a judgment unto condemnation against sinners, all who disobeyed in Adam. The righteousness of the law gave no power to the elect to fulfill its demands. Rather, serving God's redemptive plan it drove and guided the elect to seek through faith the refuge of salvation found in God's mercy and forgiveness in Christ as offered in the promises, ceremonies, and sacrifices (gospel).
In the New Testament, Christ born under the law fulfills the promise given in the Old and meets for his people the "Do this and live" demand of the law through his obedient holy life and his sacrificial death on the cross for sin. For those who believe, the law is still law but its requirements are now fulfilled and established by Jesus Christ for their salvation which they receive through faith by the hearing of this gospel. "Do we then nullify the Law through faith? May it never be! On the contrary, we establish the Law." (Rom. 3:31)
As to the so-called third use of the law for believers, it is still law. The threats of the law as given in the covenant of works no longer put the believer in jeapordy, but rather humble and inform him of what would be due his sins if not for Christ's debt payment on his behalf. In Christ this serves only to urge him toward a more thankful obedience in faith (WLC 97). Jesus Christ has fulfilled God's law for his people and yet as with all men created in God's image, they are still obliged to obey, but no longer out of fear of failure and condemnation as sinners in Adam, but willingly (though not without struggles against their ever present sin) in the assurance of God's love for them as his chosen children for whom Christ died.
So then in the New and Old covenants there is both law and gospel. But the righteousness that saves doesn't, and cannot, come to sinners through the law. As Paul writes , "the law is not of faith" (Gal. 3.12). The righteousness that saves only comes through faith in Christ (Phil. 3.9). The law cannot give that righteousness. It only comes through faith in Christ the law-keeper and sin-bearer, who has established and fulfilled it unto salvation for all who believe in him (Rom. 8.4; 10.4).