Christ fulfilled the Law and the Old Covenant (Abraham and Moses, and Noah and Adam for that matter).
Our Covenant is in Christ.
The Spirit of sonship and liberty in Christ, the Holy Spirit, is not at odds with the Old Testament law and covenants. Those parts that represent moral behavior are the same in both covenants. The ceremonial and sacrificial aspects have been fulfilled by Christ and so they are completed and no longer have to be performed.
He also perfectly fulfilled the moral law as the sinless One. The difference is that we are released from the ceremonial and sacrificial and those are done away with. Whereas we are released from the moral law as the thing which gave sin its power, according to St. Paul, when we walk in the Spirit we are fulfilling the law as we are vivified by the Spirit.
We are not under the OT law; we are under the Spirit - yet the outcome should be the same in our morals; it actually should be better!
Regarding the Sabbath. In the OT it was primarily a day for rest and family. (Hopefully in your employment you have a day or two off and get rested and time with your family). The New Covenant moved the
sabbath" from the 7th day to the 8th day, or first day, depending on how you are counting (Sunday; the 8th day being symbolic of Christian eternal rest with Christ) and put the emphasis on worship (liturgy) rather than rest from labor.