Actually, at the begining, there was only one family. And it would take generations to get to the point where the only people that weren't related to you weren't first or second cousins (since they all came form Adam and Eve.
They were both written by Moses.
The Old Testament verses set up our schedule, a life cycle if you will. It shows us how to create and maintain our work weeks, and shows the importance of that schedule. It is still important to keep a day of rest (it has been medically proven that we will live longer and healthier if we take a day off once a week from work,) but in the New Testament verses you shared, Jesus was purposely showing that caring for people is more important to him than keepng "The Law" that the Jewish leaders had set up to try to contol the people. Not a contridiction, just an advancement. Jesus doesn't say "Nevermind about that sabbath crap, we were just kidding about it."
The Sabbath is still important, but not to the point that we neglect other peoples needs.