In Matthew 5:17-19, Jesus said that he came to fulfill the law in contrast with saying that he came not to abolish it and he warned against relaxing the least part of it, so should not interpret Jesus fulfilling the law as meaning that he abolished our obligation to obey it. "To fulfill the law" means "to caused God's will (as made known by the law) to be obeyed as it should be" (NAS Greek Lexicon: pleroo), so Jesus proceeded to fulfill the law throughout the rest of Matthew 5 by correcting what the people had heard being said and by teaching how to correctly obey it as it was originally intended.
Every time someone stops at a traffic light they are completing their obligation to the law, carrying out their obligation to the law, satisfying their obligation to the law, or correctly meeting the expectation of the law, but at no point are they removing their obligation to continue to stop at traffic lights, but rather it is something that they need to keep fulfilling in perpetuity. When a husband is fulfilling his marriage vows he is correctly doing what he has vowed, not doing something that is once and done.
According to Galatians 5:14, anyone who has loved their neighbor has fulfilled the entire law, so again it refers to correctly obeying it, moreover, it refers to something that countless people have done and should continue to do in perpetuity, not to something unique that Jesus did to remove our obligation to love our neighbor.
Everything in the Law of Moses is either in regard to how to love God and our neighbor, which is why Jesus said in Matthew 22:36-40 that those are the greatest two commandments and that all of the other commandments hang on them, so the position that we should obey the greatest two commandments is also the position that we should obey the rest of the commandments that hang on them. For example, if someone thought that they just needed to love God and their neighbor, so they did not need to refrain from committing idolatry, favoritism, or kidnapping, then they would have an incomplete understanding of what it means to love God and their neighbor, and the same is true for everything else in the Law of Moses. The greatest two commandments are much easier said than done, so thankfully God gave us the rest on the Law of Moses to flesh out what it looks like to correctly obey them. Someone who was correctly obeying the greatest two commandments would be indistinguishable from someone who was correctly following the rest of the Law of Moses because they would both be following the same example that Jesus set for us to follow of how to walk in obedience to it. Moreover, in Deuteronomy 6:4-9, the way to obey the greatest commandment in the Bible is essentially by being zealous for teaching obedience to the Law of Moses. The way to obey the greatest two commandments in the OT did not change when the NT quoted them.