Biblically, blood is a symbol for life. The best-known application of this is that blood provides atonement, where one life symbolically pays the life-debt of another. However, the various covenants show a second application, where blood represents life given as a pledge of faithfulness. God ratified the covenant with Israel with blood, and those sacrificial animals gave their lives to symbolize life given as a pledge. Significantly, the blood designated as the blood of the covenant at Mt. Sinai did not come from a sin offering, but from burnt and peace offerings (Exodus 24:4-8). That covenant was sealed before the first sin offering was commanded (Exodus 29:14).
The New Covenant is also sealed with blood, but it is not sprinkled on the outside of those making the covenant, as happened with Israel. Instead, it is ingested into the innermost parts of the person. Rather than being sealed with the blood of oxen, the New Covenant is sealed with infinitely more precious blood, blood that serves as a testimony of eternal life (see I John 5:6-13), as well as a pledge of God's loyalty to those within the covenant:
Notice that God makes us complete through the blood of the covenant. Christ's blood is a pledge that God made that He will finish His extraordinary purpose - one that goes beyond forgiveness and culminates in our spiritual completion. When we are complete, then Passover will be fulfilled. But the forgiveness of sins comes through the covenant, not before it. Before the covenant, God overlooks - He passes over. When we pledge our loyalty to God through baptism, and accept His covenant after repenting, He then forgives us. When we are put into Christ, we are washed clean (see Acts 2:38; 4:12; 8:36-37; 22:16; Romans 6:3-7, 23; Colossians 2:12; Titus 3:4-7). But the cleansing blood of atonement is only available to those who accept the divine blood of the covenant.
Forgiveness is part of the covenant because we need God's forgiveness throughout the process of being made complete. Neither the Old nor the New Covenant - nor the covenant with Abraham - were preceded by atonement. Instead, God makes covenants with those whose transgressions He has passed over. It is within the covenant, then, that sin is addressed. This is why Jesus proposed the New Covenant to His disciples at that Passover observance even before He died to provide atonement the following afternoon.