In the Old Testament, David committed adultery, and the Lord God pardoned the death penalty, but he still had to bear the consequences. Is this the case today?

There is no difference between the New Testament and the Old Testament in this matter or this kind of thing.

In the Old Testament, God did not kill David immediately, because as soon as he reminded David, David realized his sin from his heart and came to God to regret his sin. God loved him and didn't give him a final judgment for his crimes, but his crimes were still not pleasing to God, so God punished him and his family. This tells us to stay away from sin, because sin has a price, and the price of sin is death. However, God has the grace to forgive sins. If we come to God and sincerely confess our sins and repent, God will forgive us for loving us. However, there are other effects and costs of crime, which we must bear and will not be changed because of God's Covenant with us.