Did Jesus cancel the entire old testament laws or just part of it?

OP: Did Jesus cancel the entire old testament or just part of it?

It appears to be Aquinas who first differentiated parts of the Mosaic Law into categories of moral, judicial, and ceremonial. See here. From this, some would assert the necessity of observing certain parts of the Mosaic Law, but not all of it.

Biblically, however, it is clear that there is only the Mosaic Law of which it is necessary to observe all of it.

If thou wilt not observe to do all the words of this law that are written in this book, that thou mayest fear this glorious and fearful name, THE LORD THY GOD; Deut 28:58

This is clearly a command to observe all 613 laws. It does not help to perfectly observe the 10 Commandments, but not the ceremonial parts like Passover, Pentecost, Booths, and more.

For whosoever shall keep the whole law, and yet offend in one point, he is guilty of all. James 2:10

For as many as are of the works of the law are under the curse: for it is written, Cursed is every one that continueth not in all things which are written in the book of the law to do them. Gal 3:10

With that in mind, we then find where Jesus fulfilled it all and in fulfilling what we cannot do, we find our righteousness in Him alone.

Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfil. Mt 5:17

The Mosaic Law is still "out there", but Christ fulfilled all of it. Thus, we find ourselves in this manner.

And by him all that believe are justified from all things, from which ye could not be justified by the law of Moses. Acts 13:39

And be found in him, not having mine own righteousness, which is of the law, but that which is through the faith of Christ, the righteousness which is of God by faith: Phil 3:9

Back to the OP

OP: Or maybe he didn't cancel and just said it isn't obligated?

Christ didn't cancel the Law, He fulfilled its requirements. We cannot do it. We cannot fulfill parts of it to any effect.

The Law is to show you your true nature. A sinner in need of salvation. Christ offers this to us. Christ who fulfilled the Law. We now live by faith.

But that no man is justified by the law in the sight of God, it is evident: for, The just shall live by faith. Gal 3:11

I suggest that we need to distinguish between God's law and the laws of Moses.

God's law is summed up by Jesus in Matthew ch22 vv37-40 as the two greatest commandments, viz our duty of love to God and our duty of love to our neighbours. He says that "all the law and the prophets" depend on these two points. The laws of Moses are what Romans ch7 v6 calls "the letter" or "the written code".

Then we can see our way clear through the mixed statements of the New Testament. The "law" which will never be abrogated, according to Jesus, needs to be understood as "God's law". The law which we no longer serve, according to Paul, is the written code of Moses. Instead, we serve "in the new life of the Spirit" (RSV). The Spirit does not leave us in anarchy but guides us through the paths of God's law. We do what is right because the Spirit tells us to do it, not because Moses tells us to do it. That is how Jesus and Paul are in agreement.

There's no need to be confused.

Rather than asking here, why not ask Jesus. This is his answer:

“Do not think that I came to destroy the Law or the Prophets. I did not come to destroy but to fulfill.
For assuredly, I say to you, till heaven and earth pass away, one jot or one tittle will by no means pass from the law till all is fulfilled.
Whoever therefore breaks one of the least of these commandments, and teaches men so, shall be called least in the kingdom of heaven; but whoever does and teaches them, he shall be called great in the kingdom of heaven.”
— Matthew 5:17–19 (NKJV)

Some of the laws in the Hebrew scriptures were given strictly for the running of the Levitical priesthood. And some of the laws were given for maintaining the civil order of the nation of Israel.

But unless you are part of the priesthood or an Israelite (you aren't, these organizations no longer exist), these laws don't apply to you now, just as they didn't apply to most people in ancient times.

But God's moral Laws, health laws, and sacred holy days and festivals are all still in effect.

What Jesus was able to "cancel" was the effect of violating God's commandments. If one realizes one's mistake, regrets and repents (changes behaviour and attitude) of it, and ask Jesus for forgiveness, Jesus will pay that penalty. But the law itself and the consequences for breaking it still exist.