Cearbhall,
there are many more features to what the NT says, and it does not say to end their lives.
1, adultery is still the same sin it always was. They are told in the NT to stop it.
2, there are changes in the administration of things when you get to the NT. It has more of message for those outside, but it also indicates changes in how to address things in the community of those who believe. The main passage you should work with is 1 Cor 5-7.
2A: The NT is not trying to tell the public family court what to do. This is part of the end of the theocracy of Israel. It does tell individuals in the public system to live righteous lives, Acts 26.
2B: The NT does not do capital punishment within its own community, but it does ask people who don't repent of what they have been openly confronted with to leave. This is in the Cor passage but also Mt 18. "Treat them as you would a tax-collector or a gentile" is perhaps the most surprising twist of all the things Jesus said, since he was found trying to reach those people. But there is a limit, just as there is a limit (actually an allowance) about remarriage. Remarriage is generally adultery; but it is not if the person remarrying was cheated. And even with that, the disciples said 'its probably better not to marry at all.' They puzzle me; Jesus does not.
2C: The NT shows divine judgement is sought. Paul sought it about the immoral person in Corinth and in Acts 5 Peter curses a couple who made an appearance of giving up substantial wealth to the church and they died.