"But now .." This can be taken either as the next logical step, or as the next temporal step. Ie either the next step in the argument, or the next step in time. In fact, both might be true, for Christ certainly did come after the Law, and it is the events around the cross that provided the basis for all that Paul was saying. Of course, it is also the next step in the argument as well.
"Righteousness of God " is a crucial term, and also one about which there is debate. Again, the two main ways of looking at the term may both be valid. People are always complaining that God is not fair. Atheists do it all the time, though how someone who supposedly does not exist can be guilty of all the crimes they accuse God of remains a mystery. Probably you and I have done it on occasion. There are certainly times when God does not seem to be acting justly. Up until this point, it did not seem that God's plans were getting very far. Adam and Eve had sinned, then Israel had repeatedly rejected and disobeyed God. Things were not looking good. The Jews thought the Law was the way that God's plans would be fulfilled, but this was not to be the case. They would be fulfilled in another way.
The other way of taking the term is in how we get into a position of righteousness. How do we gain right standing with God? Or, perhaps, how do we stay in right standing with God. The Law was a way of life, it was how they were to live in the kingdom, but if failed, or rather they failed. So now a means of attaining right standing with God was revealed in Christ that did not depend upon the Law.
Moreover, this way was testified to by the Law and the Prophets. Ie it was not new in that sense, it was not contrary to all that the Old Testament had been saying, rather it was the fulfilment of it. The Law and the Prophets were always pointing towards Christ and towards faith being the way.