Q. 1. How come we no longer hear of the Sacrament of the Last Rites? It appears to have been replaced with a new Sacrament of the Sick. Does that mean we can no longer get the Last Rites?
A. 1. The "Last Rites" and the "Sacrament of the Sick" are both the same. It is just a different name.
Prior to Vatican II, the Sacrament of the sick was called the "Last Rites." Because of its name, it was assumed that it was the Sacrament of the dying. When someone was dying, the priest was called at the last possible minute to administer the Sacrament. If the person caring for the sick waited too long, the sick person sometime died in a state of mortal sin without receiving the Sacrament of the Sick, including having his confession heard.
Now that the Last Rites is promoted as the Sacrament of the Sick, it encourages anyone who is seriously sick to call upon the priest in order to receive the Sacrament. This is in harmony with the words found in the Bible where it says, "Are any among you sick? They should call for the elders of the church and have them pray over them, anointing them with oil in the name of the Lord." [James 5:14]
So if you or someone you know is seriously sick, call upon the priest to administer the Sacrament of the Sick. It is not necessary for you or that person to be dying at that exact moment. The Sacrament of the Sick is to obtain God's healing grace. If you are seriously sick, God may heal you physically. If He does not heal you physically, at least He will heal you spiritually through the Sacrament of Confession that is part of the Sacrament of the Sick.
Be thankful to God that you no longer have to wait until you are dying to receive this precious Sacrament.