I don't think the concept of grace is one-dimensional, as it's often presented. It's not simply a state of eternal forgiveness and rightness with God that makes me feel good about myself. It's also has an element of internal empowerment.
Where the Law is a set of externally imposed rules, which our human nature and flesh cannot possibly fulfill, Grace is an internal drive and empowerment to love, which then fulfills the greatest Laws: Love God and love one another.
It's a result of the Spirit of God taking up residence in us and making us new. I think that's the heart of Romans 6:14:
"For sin shall no longer be your master, because you are not under the law (trying hard to follow the rules), but under (the empowerment of) grace."
We no longer have to be ruled over by sin, because we now have this power within us (the Holy Spirit), which is a gift of grace. We can continue to try hard and do our best, on our own, but we will be paralyzed. We are not to focus on the do/don't list, but rather on Him. Then the other stuff will begin to work itself out.
So, while grace includes rightness with God, if we stop there, we may accidentally give ourselves permission to behave however we like...which is what Paul spoke against. "Should we sin more because of grace? May it never be!" Use grace to propel you to righteousness, not just to stamp you as righteous. Is this making sense? It's hard to put into words. I guess maybe I could say it this way:
Grace is what labels you as righteous and Grace is also what causes you to actually live out righteousness.