I looked into NCT since you mentioned it. And I do not believe as they do. But understand what Paul says about keeping the Sabbath.
“One man esteemeth one day above another: another esteemeth every day alike. Let every man be fully persuaded in his own mind. He that regardeth the day, regardeth it unto the Lord; and he that regardeth not the day, to the Lord he doth not regard it. He that eateth, eateth to the Lord, for he giveth God thanks; and he that eateth not, to the Lord he eateth not, and giveth God thanks.” (Romans 14:5–6)
“Let no man therefore judge you in meat, or in drink, or in respect of an holyday, or of the new moon, or of the sabbath days: Which are a shadow of things to come; but the body is of Christ.” (Colossians 2:16–17)
Also, no mention of the Sabbath.
““Therefore I conclude that we should not cause extra difficulty for those among the Gentiles who are turning to God, but that we should write them a letter telling them to abstain from things defiled by idols and from sexual immorality and from what has been strangled and from blood.” (Acts 15:19–20)