Indeed New Testament writers and Jesus Himself taught that the New Covenant was different from the old. They also had a great deal to say about Judaizers NONE of which was good.
The New Covenant is a very real theological framework in which the person and work of Christ are understood as completing and fulfilling the OT covenant. Hebrews asserts dominant image patterns identifying the New Testament covenant as “new” (Heb 8:8, 13; 12:24; 9:15) and “better” (Heb 7:22; 8:6) than the old covenant as it is final, permanent and once-for-all, as well as being secured and mediated by Christ instead of by human priests and the sacrifices they performed. In Hebrews 12 the author uses another allusion to Abel, this time contrasting him not with Cain but with Jesus himself: “And to Jesus, the mediator of a new covenant, and to the sprinkled blood, which speaks better than the blood of Abel” (Heb 12:24 NASB).
Other New Testament passages reinforce the motifs that reach their definitive expression in Hebrews declaring the new covenant to be “new” (Lk 22:20; 1 Cor 11:25; 2 Cor 3:6). By implication the Old Testament sign of the covenant, circumcision, gives way to communion as the sign of the new covenant (1 Cor 11:25).
Jesus Himself referred to the new convenant of which Jerimiah 31:31 speaks, by which our whole natures would be changed, and God’s law written on our hearts. By his use of the words “covenant” and “forgiveness of sins” in the same breath (26:28), Jesus interprets his mission against the backdrop of Jeremiah 31:31–34. “The time is coming,” Jeremiah proclaims, when the Lord will make a “new covenant.” That time has come, according to Jesus. In his death he inaugurates the new order of salvation.
When accused by the Pharisees of breaking the Sabbath law, Jesus did not point out that he was only breaking the oral tradition. Instead, he made the astounding claim that, just as King David and the priests were ‘above the law’ in certain respects, so he was not subject to the Sabbath law, but Lord over it (Luke 6:1–11; cf. Mark 2:23–28).
The pericope in Matthew 12:1–8 reinforces the point by virtue of its position, following Jesus’ call to the weary to find rest in him rather than in the Mosaic law (11:28–30).
In the light of this, Jesus’ taking authority over the Sabbath both wrests it from the legal framework in which it previously stood and realizes the rest which God’s people were always intended to enjoy.
In all three Synoptics, the subsequent miracle is an example of what Jesus’ lordship of the Sabbath will mean in practice: people delivered from the shadow of death and restored into the unblemished image of God.
The Sabbath only features in Paul’s writings negatively. For the Galatians to observe it as if they were still under the Mosaic Law rather than New Covenant moral law would be to descend into gospel-denying slavery (Gal. 4:9–11); for the Colossians to observe it as part of a syncretistic system would be equally fatal (Col. 2:16).
For the Mosaic Law belonged to an earlier era, and since the coming of Christ it is no longer binding (Col. 2:17). Even Sabbath observance ‘for the Lord’ was tolerated only for the sake of those whose faith was weak (Rom. 14:1–12). In short, those in Christ are beyond the jurisdiction of the Old Testament Mosaic Law which has been fulfilled in Jesus.
As God’s perfect human, Jesus lived the Sabbath day for God, releasing his fellow humans from bondage and striving for salvation under the law (e.g. Mosaic Law), bringing them into a new spiritual rebirth in which the moral law of God is written on their heart, bringing them into blessing, and at the last entering Himself into God’s rest.
Ultimately, as Lord of the Sabbath, Jesus made it possible for others to follow him into that rest. This means that the Christian’s task is no longer to keep the Sabbath (Jesus has done that already) as if they were still under the Mosaic Law but rather to believe in Christ and obey Him.
In its final setting, then, the fourth commandment is no longer a commandment for God’s people, but its intent remains. The ‘law of Christ’ anticipates rest by prescribing belief, but now rest has been realized and will have its fullest expression in the gathering of genuine Christians who will reign with the Lamb for ever in the new creation (Rev. 22:3–6).
And it was all good for centuries until someone made the mistake of asking a certain William Miller to speak in their church in 1831 which resulted in him making a false prophecy and a small group of people in denial who couldn't accept the fact that the false prophecy failed weaving ridiculous heresys together to try and "prove" that it actually had happened. Rachel Oakes Preston's arrival only got them more lost and in bondage and here we are today with SDA "evangelists" maligning God's Word on CC to "save" everyone by pulling them into their gross hermeneutical error of which a modern "Sabbath heresy" became the most pronounced.
Did Paul or the Hebrew writer suggest the content of the new covenant is different from that of the old? The Bereans examined the scriptures to see if what Paul said was true. Both the Jews and the Greek believed. So did the OT say the content of the covenant would be changed? Not at all. He spoke through Jeremiah that His Law will be put in the minds and hearts of the new covenant people. It's the way how the content is put into His people that changed. And what is the content? His law.