This is an insightful question, showing me that you are already thinking about the answer to the questions you are asking.
Both Jesus and Paul have some great statements that we can use to understand the relationship between the Law of Moses and the gospel. Jesus told us in Matthew 5:17, “Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfill them. Jesus fulfilled the Law of Moses in at least four ways I can think of:
1. He was the only human who lived to adulthood who fully obeyed the Law.
2. He willingly took on himself the punishment implied in the Law, even though he did not earn such punishment.
3. He completed the Law by extending it to its limit (as seen in Matthew 5:19-48) when he said, for example, not only to love your neighbor, but also to love your enemy.
4. He completed the types, prefigures and foreshadows found in the Law (I will explain below).
So, Jesus upheld the Law, but he also replaced it with the Law of Christ (1 Corinthians 9:21). What the Law of Moses anticipated, Jesus completed. Therefore, although the Law of Moses was good, Jesus replaced the physical instructions with more spiritually-focused principles.
Paul explained that what the Law could not do (create perfection) God did through Jesus. (Romans 3:21-22). It is not that the Law was bad, but that it was a kind of tutor (Galatians 3:24), helping us to understand how sinful sin is. (Romans 5:20-21) It made sin “utterly sinful.” (Romans 7:13). In other words, the Law made us realize how truly sinful we were, but it did not fully solve the sin problem. It is the sacrifice of Jesus, not adherence to the Law of Moses that cleanses us from a guilty conscience. (Hebrews 9:14, 10:22).
So, for those who are in Christ, we are no longer subject to the Law of Moses. Although the Law was good, when we come to Christ we are no longer under obligation to Law for salvation. (Colossians 2:13-15). I believe this passage, Colossians 2:13-15 may say it better than any other. Through Christ we are not under obligation to law, including the Law of Moses. Therefore, as Paul says later, we do not have to obey things such as New Moons and Sabbaths, or other Jewish obligations. (Colossians 2:16-19) In fact, not that Christ has come, those who insist on observation of these things are definitely not doing God a favor. Paul makes this clear with the entire book of Galatians.
A bit more clarity can come from Hebrews, which describes the Old Testament Law as a “shadow” which is replaced by the “reality” in Christ. (Hebrews 10:1-4, 8:5-6 you may want to read Heb 8:1-10:19). The things in the Law, the sacrifices and the festivals are physical prefigures of the “reality,” the completion of those things, the fulfillment of those things in Christ. I have written a book on this. It is From Shadow to Reality (www.ipibooks.com).
About uncleanness, your intuition about this seems correct. The “uncleanness” in the Law of Moses is a prefigure of spiritual uncleanness in the New Testament. God calls us to be holy in Christ, but not so much through physical separation or distinctions, but by spiritual equivalents. Like I already said, quoting Galatians 3:24, the Law of Moses is a tutor, explaining through physical restrictions the mean of a more spiritual holiness. So, although there certainly was the concept of sin in the Law of Moses, as you seem to notice, the idea of sin is illustrated by the concept of uncleanness in the Law of Moses. I would not so much say that these are “different” but they are analogous, where uncleanness is a physical analogy for spiritual uncleanness (sin).
I hope this helps.