In the New Testament, there are things of the Law which the New Testament encourages, but not all of it. For example, still we are told not to murder. So, I would say Jesus has fulfilled the Law, but has not totally replaced it with our New Covenant. Also, we bring something like not murdering to a deeper level, of not hating anybody.
So, Dave, I would say technically the New Covenant does not totally replace the Old Testament.
Also, from what our Apostle Paul says in Romans 7:7, I would say the Law is still used by God for people's good, including ours.
And, by the way . . . I think of this > I don't think the Psalms are part of the New Testament, but we feed on Psalms, don't we? The Psalms can be used in a New Testament way, I guess you could say.
I think it can be risky to make a point of seeking blessings which are named in the Old Testament, because Jesus and His New Testament writers provide us with better blessings and promises, for example through Paul's prayer in Ephesians chapter three.
Jesus says to seek first the kingdom of God, and He does not say to seek the list of blessings in the Law. But we are guaranteed how our Heavenly Father will provide all He knows we need. Plus, Paul says God "gives us richly all things to enjoy," in 1 Timothy 6:17. I keep finding how He gives better than what the Old Testament guarantees, especially personally relating with God in His love and being personally guided in His peace with His creativity, while sharing personally with other Jesus family people.
But I suspect that blessings promised in the early scripture can be taken the wrong way, so someone becomes busy with getting how he or she can be blessed, and is not caring about others as much as himself or herself.
So, in case Jesus wants us to be so blessed, we need how He in us has us fulfilling His blessings the way He desires, and not only for our own purposes
We need how Jesus has us fulfilling things in His all-caring and family sharing love, in any case