It says right in what you quoted that it is when we walk according to the Spirit that the righteousness of the law is fulfilled in us. How is that not us fulfilling the righteousness of the law through our obedience? Paul says later in Romans 13:8-10 NASB that when we love others we are fulfilling the law. How is that not us fulfilling the righteousness of the law through our obedience?
You'll have to explain that one.
Which is a completely different aspect of righteousness. That is the righteousness--a legal declaration of righteousness--that we don't do for ourselves and which is gained for us by Christ. That has nothing to do with the fact that we have to walk in the Spirit (obey God) in order to fulfill the righteous requirements of the law. Not for the purpose of justification. That can only happen by Christ giving us his declaration of righteousness. We obey for the purpose of the righteousness of the law to be fulfilled in what we do. The passage you quoted above shows that does not happen until we walk according to the Spirit (love, joy, peace, patience...).
But the point is, the righteousness of the law does not get fulfilled in us until we walk in the Spirit.
If I would have thought about it better I would have realized that. No need to put a feed bag on an ox standing over the threshed out grain, right?
Besides the rabbinical add-on law allowing a man to divorce a wife, what other principle of the law of Moses does not get fulfilled in this New Covenant? Maybe there is some, but I can't think of any. The point being, the law of Moses gets fulfilled, not abolished, in this New Covenant (I know you agree it's not abolished). But surely what did happen is the law is no longer the literal covenant between God and his people. It was made obsolete in that regard, but fulfilled in this New Covenant
when we believe and obey, nonetheless.
Maybe there's a couple of laws that we don't, but I can't think of any right off hand.
It's not enough to just be in Christ to have the righteousness of the law fulfilled in us. The Romans passage you cited says we must walk in the Spirit for that to happen.
The only righteousness of the law that is fulfilled for and in us apart from our works is our legal right standing before God. That is entirely a gift given to us through our faith in the forgiveness of God in Christ.