There were many under the old covenant who came to an exceptionally God-fearing life. One cannot imagine Elijah behaving in an improper way with the opposite sex or John the Baptist running after money or Moses seeking the honour of the world. So the mere fact that we have been freed from such matters does not mean that we have entered into New Covenant.
If a weekly paper on the Scriptures had been published in old covenant times, many good things would have been found in such a paper. But the underlying note in it would have been, "Thou shalt..." and "Thou shalt not..."
Under the new covenant it is different. It is not just that the standard is higher. The spirit is also different. Consider the words of Jesus in Mathew 5,6 and 7. The old covenant had commanded "Thou shalt not kill... Thou shalt not commit adultery..." etc. But when Jesus spoke about the spirit behind these commandments in Mathew 5:21,22,27,28, he did not use the words "Thou shalt not..", but merely explained that in the Kingdom of God, anger was equivalent to murder and sexually dirty thoughts were equivalent to adultery. It is important to note this difference in spirit.
Under the old covenant, it was compulsory to give 10% of one's income to God. But under the new covenant, there is no command indicating what percentage is to be given to God. On the other hand, it says, "Let each one do just as he has purposed in his heart, not grudgingly or under compulsion; for God loves a cheerful giver" (2 Cor 9:7). This is to be the spirit of obedience under the new covenant, to all of God's commandments: NOT UNDER COMPULSION BUT CHEERFULLY.
Under the new covenant, the spirit in which we obey is what God values - not just the obedience. There is difference between a slave who does what he is commanded, because he has no choice and a grown- up son who obeys his father cheerfully out of love. Galatians 4:7 states that, that is the difference between the old and the new covenants. Employees will work in an office, for 8 hours a day for a salary. A son however, will work in his father's office, if necessary for 16 hours a day, without any pay. God is looking for SONS now, not for employees (servant). The old covenant has been abolished (Heb 8:13)
It is possible, for people to keep the new covenant commandments in the spirit of the old covenant. Such people usually develop into new covenant Pharisees. They can then hinder the building of the church, for their soulish zeal can come forth with many "Thou shalt"s that they seek to impose on others to conform to the outward form in the church.