I appreciate your faith and your respect for the inspiration of the Bible. I agree with your general premise that there are no bona fide contradictions in the Scripture. I think it is a bit too strong to, with a broad brush, declare all proposed examples of potential contradiction “intellectually dishonest,” or even taken out of context. I do agree that many proposed contradictions come from critics who are not particularly interested in a reasoned and balanced discussion of the issues and who could be described as “intellectually dishonest,” However, there are a good number of legitimate questions about the consistency of the Bible which open-minded honest people do bring up, and I do not want to impune the motives all those who claim that there are contradictions in the Bible. In fact, your question is one of these legitimate questions about the consistency of the Bible!
Now, about your question, this brings up the topic of the relationship between the Old and the New Testament, and more specifically what are the implications that the giving of a New Covenant by God have on the relevance of the former covenant. Jesus said in Matthew 5:17 that he did not come to abolish the Law (presumably including the Mosaic covenant), but that he came to fulfill it. He goes on to say that “It was said to you of old…” In these statements he explains how the New Covenant replaces the Old Covenant. Paul explained the purpose of the Law of Moses, which is to “make sin utterly sinful” (Romans 7:13). He also said that “The law was brought in so that trespass might increase.” (Romans 5:20). What he is saying here is that the Law of Moses had the effect of helping us to understand how sinful we truly are, as we were completely unable to fuflill the requirements of the Law. This made us feel our deep need for a Savior, which is provided in the New Covenant, as I am sure you are aware! Paul also said that “the Law [of Moses] was our guardian until Christ Jesus came that we might be justified by faith.” (Galatians 3:24). In other words, the purpose of the Law of Moses, and to some extent even of the entire Old Testament, was to prepare us for the full revelation of his will through the person, life and ministry of Jesus Christ.
This being the case, we, as disciples of Jesus, are not subject to the Law of Moses. Instead, we are under the “law of Christ.” (Galatians 6:2). In Colossians 2:14 we are told, “having canceled the charge of our legal indebtedness, which stood against us and stood opposed to us, he has taken it away, nailing it to the cross.” This passage is not simply referring to the Law of Moses, but law in general, but the point is that we are no longer subject to the requirements of a legalistic system, but are under Christ, in whom “the only thing that matters is faith, expressing itself in love.” (Gal 5:6)
This is a somewhat long way of saying that, as Christians, we are not required to follow the Old Testament laws. We are not required to sacrifice animals, or to obey the Sabbath (Coll 2:16-17), or the observe the Jewish holy days. It is in this sense that the former covenant was, in tbe words of the Hebrew writer, “old and fading and will soon disappear.” In fact, soon after Hebrews was written, the temple in Jerusalem, which was a mere hollow shell after the death and resurrection of Jesus, was destroyed by the Romans in AD 70. This is the meaning of the texts you mention. All Scripture is God-breathed, and is useful, but the Old Testament has a different use than the New Testament. For example, Paul tells us in 1 Corinthians 10:6 that the events of the Old Testament are “examples for us, to keep us from setting our hearts on evil things as they did.” Or as Pail says in Romans 15:4, “For everything that was written in the past [ie in the Old Testament] was written to teach us, so that therough the encouragement and endurance taught in the Scriptures, we might have hope.” God teaches us an incredible amount through the Old Testament about how to live by faith, to live in submission to him, the horrible consequences of sin and rebellion and idolatry and so much more. What we do NOT take from the Old Testament is the laws which were given specifically to the Jews.
Some say, well, what about the Ten Commandments? The answer is that these are part of the Law of Moses. Nearly all of these commandments are implied by the New Testament. We do not bear false testimonly and covet our neighbor’s things because we learn from Christ not to do these things. We learn not to kill or to commit adultery or to steal from Jesus. However, the third of the ten commandments, the one to follow a Sabbath does not apply to us, as Paul said directly in Colossians 2:16. If there is a law found in the Old Testament which is also implied by the life and teaching of Jesus, then we follow that law, not because it was given to the Jews at Sinai, but because we are following Christ. I agree that there are (to quote you) “certain things that still apply” but that is because we learn these “laws” from Christ. There is no contradiction here. We are under a different covenant. As Christians we have a massive amount to learn from the Old Testament about God and his relationship with us, but we do not borrow laws from Sinai which are not implied by what was taught or demonstrated by Christ. For example, Jesus specifically said that the kosher laws regarding food in the Jewish system is no longer in force for Christians. We find this in Mark 7:19. “In saying this, Jesus declared all foods clean.” This principle that Jesus suspends the Laws of Moses, not by abolishing the, but by fulfilling them and replacing them is the answer to your question. The laws given to the Jews at Sinai are obsolete because Jesus replaced them. However, the Old Testament is certainly not obsolete!!! It is amazing, inspired Scripture which tells us so much about God, but it is not a source of “laws” or rules for Christian living. We are under the law of Christ.