Answer: Apostle Paul agrees with your overall assessment when he stated that the Eternal's laws, found in the Old Testament, was not only good, but holy and just (Romans 7:7,12).
There are three parts of the Old Testament law of God. The first is the spiritual law, codified in the Ten Commandments, which reveals how a person should conduct their lives and how they should treat their fellow man. The second is the "civic" or government commands that dealt with the running of civil society with God as the head of that society and literally dwelling with the children of Israel.
The third part is the Levitical or ceremonial commands that included all the animal sacrifices and other ceremonies associated with the Levitical priests. It should be noted that the spiritual laws abide forever.
Because we do not live under a civil government headed by God, like Old Testament Israel, we cannot strictly enforce all the civil parts of the Law. That said, the judicial systems of many countries contain (at least in principle) parts of it.
A change of priesthood
As for the Levitical laws, Paul explains in Hebrews chapters 7 - 10 that we have had a change of priests under the New Covenant. This change does away with all human priests and recognizes only Jesus as our High Priest (see especially Hebrews 7:12 - 16, 23 - 28, 10:12 - 14).
Paul directly answers your question about whether we should keep other parts of the law such as sacrifices and sin offerings in Hebrews 7:12. According to him, we are not obliged to keep any of them associated with the Levitical priesthood such as those that are ceremonial.
A significant portion of Old Testament law is not only still valid today, but will certainly be in force after Jesus' Second Coming. The Bible states that at that time ALL the nations will flock to Jerusalem to be taught his laws and commandments (Isaiah 2:2 - 3).
Though in force, only a very few converted people today keep God's annual Feast Days let alone most or all of his laws. This will change when Christ returns, however, as he will command all the nations to keep his Feast days which he delineated in the Old Testament. These special days include the Feast of Tabernacles (Zechariah 14:16 - 19). If nations choose not to observe these days they will be swiftly punished.
If any nation refuses to go and worship the Lord Almighty as king, then rain will not fall on their land. If the Egyptians refuse to celebrate the Festival of Shelters (Feast of Tabernacles), then they will be struck by the same disease that the Lord will send on EVERY NATION THAT REFUSES TO GO.
This will be the punishment that will fall on Egypt and ON ALL THE OTHER NATIONS . . . (Zechariah 14:17 - 19).