There are two types of laws, not three or four.

God’s Law is often broken into three or four categories i.e., Moral, Civil, Ceremonial, and Dietary/Hygiene, if the latter is not included as part of the ceremonial. This is in spite of the fact that scripture makes no such delineation. Scripture divides law into two and only two categories: Absolute and Covenantal. As for absolute law, we read, for example

¹⁴ For when Gentiles, who do not have the law, by nature do what the law requires, they are a law to themselves, even though they do not have the law. ¹⁵ They show that the work of the law is written on their hearts, while their conscience also bears witness, and their conflicting thoughts accuse or even excuse them ¹⁶ on that day when, according to my gospel, God judges the secrets of men by Christ Jesus. (Rom 2:14-16)

There is, as I read this and other passages, an absolute law, call it a moral compass, written on the hearts of all people. Now, some have argued that this law written on the hearts of all people is in fact the Decalogue. However, there is no scriptural or anecdotal evidence for this. To be sure, whatever the law is written on the hearts of all it contains some overlap with the Ten Commandments. For example, the overwhelming majority of humanity has a moral objection to murder. However, there is no similar universal morality in evidence in regard to keeping the Sabbath. Unbelievers do not feel a moral twinge when they do not obey the 4th commandment. Indeed, prior to the Exodus there is no evidence in scripture that any human being of any religion, nation, race, or tribe from creation onward felt a moral obligation to obey the Sabbath.

To support the argument that it is the Decalogue written on the hearts of all people, some point to this passage which occurs right after the commandments are recorded:

²⁹ Oh that they had such a heart as this always, to fear me and to keep all my commandments, that it might go well with them and with their descendants forever! (Deut 5:29)

But this actually means just the opposite. It is a lament that the Decalogue is not on the hearts of all men. If they were it would appear to be an upgrade, given that their presence is described in glorious terms.

That there is absolute law written on the hearts of all people certainly does not require that the entirety of God’s absolute law be on every heart. It means only that what is on every heart is a subset of God's absolute law.

The other type of law, the only other type of law, is covenantal law. This is law that applies to the period that a covenant is in force, and ceases to apply afterwards. There may be another covenant that supersedes, and its covenant law may contain law that overlaps with the prior, but a covenantal law is in effect only because it is part of the law of the current covenant.

There were laws regarding the proper practice animal sacrifices in the Mosaic covenant. They are no longer in effect not because they were ceremonial laws per se, but because they are not part of the covenantal law of the New Covenant. (And in fact their practice, once legal and moral, would now be so radically against the current covenantal law as to qualify as an abomination.)

The law-giver for the New Covenant is Christ. Any (or all) of the Ten Commandments that still bind us (not going to discuss that here) are binding because they are laws under the current covenant, the New Covenant. It is not because they are written on the hearts of all men (some might be, some clearly not) and it is not because they were exempt from the set of laws that ceased to be binding with the termination of the Mosaic  or old covenant. If they are binding, it is for one reason only: they were renewed under the administration of the New Covenant.

This view is perhaps best supported in this passage:

²⁰ To the Jews I became as a Jew, in order to win Jews. To those under the law I became as one under the law (though not being myself under the law) that I might win those under the law. ²¹ To those outside the law I became as one outside the law (not being outside the law of God but under the law of Christ) that I might win those outside the law. (1 Cor 9:20-21)

I cannot read this passage without concluding that Paul is reffering to defunct laws of the old covenant and different laws of the new covenant, the latter of which he describes as the law of Christ. And how, being something as of the world's leading expert on both, the old/dead and the new/alive, he could use that knowledge to his advantage when evangelizing.