"What, do ye suppose that mercy can rob justice? I say unto you, Nay; not one whit. If so, God would cease to be God." (Book of Mormon, Alma, 42:25)
Mercy does not rob justice. However, justice is not a monster that God must appease. Gods justice is God doing exactly as He has said and Gods mercy is God doing exactly as He has said.
Blessings are predicated on obedience. The Atonement of Jesus Christ promises that we will repent the consequences of past sins will not follow us forever. Instead, through Christ's blood, we can begin to repent and begin to obey and begin to be blessed. However, some consequences do follow according to the law of the harvest (we grow what we sow). But what we cannot do is to continue in sin and expect to be blessed.
We cannot claim to accept Christ as our personal Savior and expect to be saved in our sins or expect to be blessed while continuing in our same sinful path without any change or repentance. Accepting Christ won't cause us to completely loose our sinful natires and become perfect overnight, but, as we draw nearer to Christ, we will begin to change, begin to be delivered from the consequences of past sin, and begin to be blessed.
As we accept Christ, we become born again. An individual who is born again is not immediately perfect, but desires to cast off the natural man and to submit to the enticings of the Holy Spirit. They desire that the Lord God who created them should rule and reign over them. They seek for the day when they eventually will loose all desire for sin and seek no more to do evil but to do good continually. But the realization of these righteous desires and the realization of the resultant blessings is a life-long process requiring keeping the faith, finishing the course, and enduring to the end. This is how the Book of Mormon teaches us how Jesus Christ saves us from our sins and not in our sins.