Many people do still follow Biblical law, some with better ability and some with less, but their intent is to follow it.
The Mosaic law of the Bible can't be understood out of the context of Jewish Oral Law which is in the Talmud. Essentially, as a Jewish Court of Law is always required to entertain doubt in the case of some one liable for death penalty or otherwise heavy physical or spiritual penalty, the death penalty is close to never carried out.
One recorded case of death penalty being carried out is on the son of Judah ha Nasi (the head of the Jewish Sanhedrin and a Cohen,jewish priest, who lead the Jewish people after the Roman destruction of the second Temple and the sack of Jerusalem) and later it was indeed learned that there had been false witnesses who testified he had comitted heretical acts to the Sanhedrin. The act of false witness was discovered too late, only after he was hung, when one false witness repented and gave testimony against himself. This is a midrash (tradition of the law sometimes allegorical, in this case one of case law), which illustrates why a Jewish court never carries out the death penalty even in the case of a transgression (such as adultery or failure to keep the Sabbath (actually the charge for death is defiling the Sabbath in public, rather than mere failure to keep Sabbath).
Adultery is carrying out a sex act (whether you are a man or a woman) with a married man or woman who is presumed to be unavailable (that is you do not have permission from the court, the other spouses or spouse etc). to have relations with that person, and there is no Jewish court that will carry out the death penalty, except in the sense that being embarrassed publicly in the Talmud (the Jewish Oral Law) is considered a type of death.
Christians frequently lay claim to Jesus telling all that they may not change one jot or tittle of the law, yet apparently have excused themselves of at least attempting to carry out much of it (when it comes to the laws of purity such as kosher etc.) but do try to carry out proper behavior towards other human beings (and in Jewish thought the sins you carry out against God through lack of purity are between you and God are between you and God and asking God to forgive you and then make an honest attempt to carry out those laws that you can does get you forgiveness), while sins against human beings who are frail and easily harmed must be forgiven by the person you have sinned against after you have made an honest attempt at making amends and righting the wrong done as best you can, before they can be forgiven by God. Thus the Christians who truly practice the teachings as they know them, are at least avoiding the more grevious class of sin - that against humans.
The only other known case of a Jewish court carrying out a death penalty was against Eichmann and there were hundreds plus of witnesses against him and his crimes were among the most heinous type known to humanity (genocide). (And this is not saying he and Hitler carried out the only genocide, the Pol Pot regime carried out genocide, the Turks killed one million Armenians, look at Sudan, Congo, the European conquest of the Americas etc.)