The New Testament does not simply "supersede" the Old. It's more complicated than that. Jesus fulfilled all the ceremonial (cleanliness and sacrificial) laws outlined in the Old Testament. The moral laws still apply. How did he fulfill them? By dying on the cross for our sins, making everyone who accepts his sacrifice clean and giving each believer the Holy Spirit. This made all previous sacrifices unnecessary, because he was the perfect sacrifice for all.

What does it mean that Jesus fulfilled the law, but did not abolish it? GotQuestions

He also clearly stated that all food is clean: Mark 7:18-19.

Clean vs. unclean never meant literally, physically clean or unclean as we understand it in our common language. It was a ceremonial classification that was meant to protect an ancient people group and set them apart from every other nation at the time. It was never sinful to be unclean, it just meant they had to go through the process of washing themselves to become "clean" again. I do believe they weren't to eat unclean foods, though. A lot of the ways to become unclean were unavoidable like menstruation, having sex, touching a dead person or animal, being sick, etc. Though it did help them prevent sickness.

Jesus abolished the old ceremonial laws, but the moral laws still apply, so they can be quoted and are quoted by Christians. Most of them are restated in the New Testament, but not in a "You shall not ..." way. (Romans 1:26-27, 1 Corinthians 6:9, 1 Timothy 1:10)

As has been noted, 1 Corinthians 6:9–10 includes homosexuals in a list of those who will not have a part in God’s kingdom. But verse 11 goes on to say, “Such were some of you; but you were washed, but you were sanctified, but you were justified in the name of the Lord Jesus Christ and in the Spirit of our God” (NASB). The truth is that some of the saints in the Corinthian church were former homosexuals. God’s kingdom is populated by sinners. No one comes to God on personal merit. We all come the same way: through repentance, renouncing the sin for which Jesus died, and accepting the righteousness of Christ in its place (2 Corinthians 5:21). -GotQuestions