Evangelism is telling people they need Jesus and how much He loves them. He’s the only One who can forgive sin. This is fulfilling our mandate to preach the Gospel. We preach it because we know that many people seek spiritual satisfaction in false practices. We know they are hungry and that only the Bread of Life truly satisfies.
But we should not dismiss the OT. The Law must precede the Gospel. It must come first and kill the old nature so that the Gospel can make them alive and new. The Law must convince people of their sins and by doing so they will desire salvation. So in evangelism we must remind them of the 10 Commandments (Rom.10:5). As you said Jesus has not abolished but fulfilled the Law. If a doctor told us to take some pills for a week, but didn’t tell us why, would we take them? In much the same way, we can’t people to "swallow" the Gospel by simply saying "You need Jesus!" We need firm foundations when evangelizing (Mt.22:36-40).
The purpose of the Law was to show mankind the desperate situation we’re in by revealing our sin (Rom.7:7). Then because we know we’ve sinned against God, the Law then reveals that we are under divine wrath (Rom.4:15) and physical and spiritual death (1 Cor.15:56). That’s how the prophets spoke of the Law that people have violated the will of God and that they are going to face the terrible judgement of God on the Day of Wrath (Rom.2:5). If we don’t share this when explaining the Gospel we’re not presenting the real reason for the Gospel and this can hinder people from coming to Christ. Also remember Galatians 3:24: "God presented the Law to become our tutor to lead us to Christ, that we may be justified by faith."
Because of the harshness and truth of the Law we’re broken before God and recognise we can do nothing to please Him (Is.64:6). It breaks us so that we are found hopeless inside and we turn to another to deliver us from the wrath of God. Therefore the only option is to come to the Cross. The Law "pushes" us toward the One who forgive sins. This is why Jesus came. This is the Gospel. Jesus died on the cross, the perfect sacrifice, to avert the wrath of God from sinners. This is what it means to be saved (Rom.5:9).
So when we preach the Gospel we must make sure we speak of the Law too. It can break the heart open so the seeds of the Gospel can be sown. That’s the foundation we need. The Law is our moral and ethical guide and through Jesus we can understand the true and full intentions of it.