There's an awful lot of confusion about the Old Testament in Christianity. Anti-Christians and liberal Christians are especially confused, spitting out cliches like "why do you eat shellfish?" and other things.
This really is a case of making a mountain out of a mole hill. Generally speaking, educated people like to distinguish between moral law, civil law and ceremonial law. This is good, but really, we only need two distinctions: Moral and covenantal.
Effectively, while God generally deals with all humanity in one way, He makes specific deals with specific people. Murder, for example, is always wrong for all people, for all times. This was true before Moses, this was true after Moses. This is called moral law, and in Romans 2:14-15, we actually see that even gentiles (non-Jews) are at the very least subconsciously aware of moral law. Which, by the way, is why even atheists can have some semblance of a moral compass. They may chant "good without God", but they're using their God-made conscience to understand that good is even a thing.
By contrast, things like dietary requirements vary. For Adam, only plants were permissible. For Noah, anything was permissible, as long as the blood had been removed first. But then God made a covenant with Israel, and so Jews, and only Jews, were subjected to various food laws. But now, we are told that all foods are permissible, and that to say otherwise is actually called the doctrine of devils (1 Timothy 4:1-3).
That being said, moral and covenantal law are not 100% separate. Aside from the overlying moral law that God must be obeyed, God sometimes includes specific moral laws in a specific covenant. Again using murder as an example, Mosaic law does include a command not to murder. This is because it also contains instructions on how to deal with murderers. See, the Mosaic covenant was tied to a piece of land: Israel. Israel was intended to be a semi-Theocratic nation (I say semi because although God gave the main laws, He wasn't personally sitting on the throne), and so the law of Moses was, quite literally, the law of Moses. Kings were required to know and enforce it throughout Israel.
Because of all this, it makes sense that Christians do not obey Moses' law. The covenant God made with us is distinct in many ways, not even tied to a piece of land or a quasi-Theocratic political system, but rather tied to God's eternal Kingdom.
Effectively, moral law is a result of God's character. When we break a moral law, we offend God, because we have done something contrary to Him. It is this that separates us from Him. Covenantal law has a variety of purposes, but the purpose of the Mosaic covenant specifically is to guide and protect Israel, both creating the conditions for the birth of the Messiah, and allowing us to recognise Him. Mosaic law may not apply to us as it did to pre-Christian Jews, but it does apply to us in a different way. Without Moses, there would be no Jesus, and thus no forgiveness for sins.
All of this could be compared to our own laws. There is a distinction between criminal and civil law. Criminal law concerns that which no one can legally do within a jurisdiction. Once again, murder is an excellent example. No one has the legal right to randomly brick someone in the head until they die. This action brings with it a lengthy, likely lifelong prison sentence. By contrast, there are individual contracts which, while legally binding, are only binding upon those who make them. I can't walk up to your boss and demand your salary, but if your boss isn't paying you, you have the right to take him to court.
You see, then, the distinction between moral and covenantal law in the Bible. Christians aren't cherry picking, nor are we being hypocritical. We are simply applying each commandment as it was contextually intended to be applied.