Greetings stunnedbygrace
The OT lambs as you say were a shadow of the Messiah to come...Those lambs were innocent having no sin, and so was Jesus innocent having no sin...The life is in the blood and this is what God required for atonement..What is atonement? Atonement, a beautiful word created by William Tyndale in the English, he created many English words by combing the Hebrew, an the Greek with middle English, forming modern English. The man was a genius in languages, he probably accomplished more than the 47 KJB scholars combined who came about 70 years after him.."Passover" is another beautiful word developed by Tyndale in the English........
Anyway, the blood did not pay for sins, it provided an atoning work that God required so HE could dispense HIS mercy and forgive sins...Animals could only cover sin, they could not take away sin, Heb.10-4,11, however the Lamb of God, the Messiah came to take away sins, as prophesied He would in the mist of Daniels 70th week when HE was cut off for our transgressions at the cross, and take away sins, Daniel 9:24-27...John the baptizer said, behold the Lamb of God which TAKETH AWAY sins, John 1-29....Jesus blood has the power to remove sin from our life, (HE shall save his people from their sin, Matt. 1-21...For this is My COVENANT with them, when I shall TAKE AWAY their sins, Romans 11-27......And ye know that he was manifested totake away our sins; and in him there is no sin, John 1-35... If His blood could only cover sin it would have no more power than the OT lamb sacrifices....
The blood of Christ does not cover sin, it remits sin if one has truly repented and is not out to get one over on God, by returning to their vomit...God sees the heart in repentance and looks for purity. He looks to see if their is any guile left therein; that is what is meant when the text says God searches the hearts of men..He searches the heart for guile and to see if the heart is sincere......
Would be most pleased to answer any more questions concerning the atonement that you might have...Its probably the most misunderstood doctrine in the NT......................be blessed always in you searching