There's no reason different NT writers can't use the same term in different ways. There's also no reason other than conservative Protestant ideology that they can't disagree.
James probably means "justify" to mean "show to be righteous," and he probably means "righteous" to have the usual Biblical meaning of living a life that God approves.
When Paul uses justification and righteousness he's talking about what makes someone a Christian, not how they live as a Christian. He agrees with James that to live as a Christian you have to live right. N T Wright believes that justify for Paul means "show to be righteous," but it's righteous in the sense of being a proper member of the covenant people. I think justify sometimes refers to how God puts someone into that proper status, but that's probably not relevant to this thread.
So technically James 1:21 ff doesn't contradict Paul. Indeed Paul agrees that proper behavior is necessary, largely in other books such as 1 Cor.
But more broadly, we know that Paul and James disagree, because Paul says so. What's unusual in Paul's theology is that he treats our status as righteous before God as based on our faith, and our lives as Christians as a consequence of that.
Even 1 Cor 6:9, which is typically quoted to support a kind of Protestant version of mortal sin, retains this distinction. He says that the Corinthians have already been washed and justified. They are now required to live up to that.
There's no sign of that distinction in James, and based on his disagreement with Paul it's very unlikely that he actually would agree with it. This doesn't mean that James is useless, just that he's not the first place to look to see the theological implications of grace.