Because there can not be such thing as an objective proof, especially when the subject matter is as personal as one's connection to the divine, this answer can not be any more than a mere opinion. Therefore it seems safe to begin with "I think..."
I think the answer to your question lies in the phrase from the Old Testament, quoted by Jesus in the New Testament: "I desire love not sacrifice". What is a sacrifice in this context? It is a price to pay to purchase goods from God, that is a specified algorithm aimed at obtaining something from God. So it seems laws and sacrifice are the same thing in this respect, if their aim is your connection to Him. In other words if you are trying to purchase God's favor, or manipulate him through rules, payment, or incantations, you're not thinking your actions all the way through. From this perspective, one of salvation or connection, none of the laws ever applied to neither Christians nor the Jews.
So the reason those laws are presented in the scripture must be something other than salvation. In addition, interpreting those laws never was a straightforward affair. Moses, who delivered "thou shall not kill", led Israelites to kill the inhabitants of the Holy Land. Many Jews, living abroad, could not stone other Jews for adultery. How did they decide what scriptural laws were applicable, and when, was as much of a problem then, as it is now, and so one needs to know their purpose for them to have benefit.
It seems each law has to be considered individually, depending on the situation one's in. Can you afford to kill your own animals in a fantastic ritual? Maybe you should, it would certainly be more artistic and beautiful than mass machine slaughter at the meat factory, having no idea who touched your meat before you bought it at the supermarket. Should you stone people for adultery? If you live in a country that permits it, then you have God's permission to cleanse that country of it, of course accepting full well that you should present yourself and your family and children for stoning, for stonable offenses, should you or they commit such. Should you celebrate the prescribed holidays? Couldn't hurt, considering that people have no idea how to celebrate holidays these days, but as with all others your salvation does not depend on them.
The only things that were truly cancelled in the New Testament (by the apostles, not Jesus, nonetheless) was the circumcision, which wasn't quite a law, but more of a mark of an era, and the dietary restrictions, which seem very minor of the laws, and although I do not have an understanding of why they were given with the rest, I am inclined to believe that the point of them was prevention of some diseases that were common in those forbidden animals in the ancient times.
So nothing is changed, as long as your aims are not to manipulate God, following the rules of the Old Testament, as far as they are legal in your country, can improve your life, but discretion should be practiced and there are far more important things to be mostly concerned about, that the former should naturally flow from. Perhaps the phrase "law written on their hearts" pertains to that.