Also they lean really heavy on the old testament even when it contradicts the teaching of Jesus.

No. it actually doesn't. Jesus taught a strict adherence to the Law of Moses. Specifically, the written Law. he states this incredibly clearly in Mathew 5:17-20:

^17^“Do not think that I have come to abolish the Law or the Prophets; I have come not to abolish but to fulfill. ^18^ For truly I tell you, until heaven and earth pass away, not one letter, not one stroke of a letter, will pass from the law until all is accomplished. ^19^ Therefore, whoever breaks one of the least of these commandments and teaches others to do the same will be called least in the kingdom of heaven, but whoever does them and teaches them will be called great in the kingdom of heaven. ^20^ For I tell you, unless your righteousness exceeds that of the scribes and Pharisees, you will never enter the kingdom of heaven.

Unless you care to completely ignore the words we ascribe directly to Jesus himself, of course. Which is what most Christians will do. They will say something to effect of "the law is fulfilled therefor he abolished the old covenant and we have a new one", but that's not what jesus is saying at all. What he's saying he came to fulfill is the messianic prophecies. The law itself is eternal, and will persist until the end of times.

There's also a general lack of understanding about the cultural context. Romans were severely oppressing the jewish people. the kingdom of Israel itself was a client-state that had been conquered and subjected to roman rule; and such like. People were angsty. And their traditions taught that they were the chosen people. And in the past, when they got roflstomped by much larger nations, they explained it as their god being pissed at them... because they stopped following the Law that god laid down. (or rather, Moses laid down... details.)

and so god punished them with getting conquered and war. the most famous of these are the Babylonian exile and such like.

So we come to the Roman occupation/oppression/etc of the period and there were a lot of people that were upset that the pharisees followed something called The Law of the Elders, or also, The Oral tradition (Which was the law of moses as passed down orally,), which was a softened version that took off the rough edges of the written law... because frankly, it was straight up awful.

It's a fundamental misunderstanding that Jesus was somehow 'more progressive' than the pharisees. he was actually a fundamentalist. you can see this distinction in when he's beefing with them. For example, Mark 7:

^1^Now when the Pharisees and some of the scribes who had come from Jerusalem gathered around him, ^2^ they noticed that some of his disciples were eating with defiled hands, that is, without washing them. ^3^ (For the Pharisees, and all the Jews, do not eat unless they wash their hands, thus observing the tradition of the elders, ^4^ and they do not eat anything from the market unless they wash, and there are also many other traditions that they observe: the washing of cups and pots and bronze kettles and beds.) ^5^ So the Pharisees and the scribes asked him, “Why do your disciples not walk according to the tradition of the elders but eat with defiled hands?” ^6^ He said to them, “Isaiah prophesied rightly about you hypocrites, as it is written,

‘This people honors me with their lips, but their hearts are far from me; ^7^ in vain do they worship me, teaching human precepts as doctrines.’

^8^ “You abandon the commandment of God and hold to human tradition.”

^9^ Then he said to them, “You have a fine way of rejecting the commandment of God in order to keep your tradition! ^10^ For Moses said, ‘Honor your father and your mother,’ and, ‘Whoever speaks evil of father or mother must surely die.’ ^11^ But you say that if anyone tells father or mother, ‘Whatever support you might have had from me is Corban’ (that is, an offering to God), ^12^ then you no longer permit doing anything for a father or mother, ^13^ thus nullifying the word of God through your tradition that you have handed on. And you do many things like this.”

The first part, the accusation by the Pharisees is not part of the written law. It comes from the Oral Tradition. The Pharisees were upset that the disciples were breaking that, and Jesus was like "but you don't even follow the Law of Moses!"... and then cites a law that requires stoning unruly children. Specifically, exodus 21:17 and Leviticus 20:9, which say:

Whoever curses father or mother shall be put to death. and
All who curse father or mother shall be put to death; having cursed father or mother, their bloodguilt is upon them.

The original written law was something wishy washy, so rabbinic tradition (law of the elders) made it less wishy washy, until there very few justifications for actually stoning children. It started simply as bitching and moaning about your parents, as Jesus took it, and they limited the whole stoning-thing to things much, much more egregious (to the point that the standard was basically never met.)

it's important to note that Jesus and people at that time made very sharp distinctions between classes of people. Jesus wasn't talking about, for example, slaves. Jesus literally had more to say about paying taxes to oppressors than slavery. Not because he didn't encounter slavery (that was probably a daily occurrence, and Jerusalem would have had a thriving slave market,) but because he was fundamentally okay with it. So when jesus talks about 'the least of these', it's important to remember he's talking about his brothers and sisters- presumably his followers. not the slaves who were the literal least in society. to his conception, they weren't people- they were property.