The first Covenant Gen 2 is "obey and Live". Gal 3 confirms this.
Adam and Eve sinned so at that very moment the New Covenant , grace became the rule, else they would have died then and there.
Well, he was condemned to work by the sweat of his brow, effectively missing out on God’s rest.
Christ did not "fulfill the command to not take God's name in vain by dying". Dying is not fulfilling moral law, rather it is paying the debt owed for violation of moral law.
The contract broken, called for death.
Even so, it is still a sin to take God's name in vain.
3For what the law could not do, in that it was weak through the flesh, God sending his own Son in the likeness of sinful flesh, and for sin, condemned sin in the flesh:
4That the righteousness of the law might be fulfilled in us, who walk not after the flesh, but after the Spirit.
The covenant with Abraham had two parties, God and man. By becoming man and dying the death, the covenant is done away with completely because God no longer remains a party.
7So I became the shepherd of the flock doomed to be slaughtered by the sheep traders. And I took two staffs, one I named Favor, the other I named Union. And I tended the sheep.
8In one month I destroyed the three shepherds. But I became impatient with them, and they also detested me.
9So I said, “I will not be your shepherd. What is to die, let it die. What is to be destroyed, let it be destroyed. And let those who are left devour the flesh of one another.”
10And I took my staff Favor, and I broke it, annulling the covenant that I had made with all the peoples.
11So it was annulled on that day, and the sheep traders, who were watching me, knew that it was the word of the LORD.
12Then I said to them, “If it seems good to you, give me my wages; but if not, keep them.” And they weighed out as my wages thirty pieces of silver.
13Then the LORD said to me, “Throw it to the potter”—the lordly price at which I was priced by them. So I took the thirty pieces of silver and threw them into the house of the LORD, to the potter.
14Then I broke my second staff Union, annulling the brotherhood between Judah and Israel.
There remains then a Sabbath.