All ways of speaking of our salvation are analogies at best. There is a reality that exists about our salvation and our relationship to God, but human language can never fully articulate it. So we compare our salvation to other human experiences, and some approximate that reality better than others.
"Forgiveness" relates to a debt. If I owe you money, you can either collect what you owe, or you can drop your claim to it. If you drop your claim to my debt, then my debt is forgiven. Salvation is kind of like that, with our "debt" being our sin.
"Ransom" relates to a kidnapping. If my enemy has kidnapped my child, then I will pay that enemy whatever he demands in order to get my child back. Salvation is kind of like that, with the "enemy" being Death or Satan, and the "ransom" being Christ's blood.
"Redemption" relates to finding something lost. If I lose something, I'm going to look for it until I find it. If I find out where it is, I will go to where it is and claim - or "redeem" - it. Salvation is kind of like that.
So we have to first remember that language such as "forgiveness" is not so much a concrete reality as much as it is a human way to trying to express an inexpressible reality. But forgiveness does not necessitate a payment. In fact, if a payment were made - even if it were not the indebted party - then there is actually no forgiveness at all. If God requires payment, even if that payment is made by Christ, then our sin is not forgiven; it's just paid by someone else.
Christ's blood didn't have any kind of special properties, and it wasn't the act of the blood exiting His body that was significant. It was His entire act of voluntarily laying down His life to the point of death. Christ did in fact shed His blood as part of the way He was murdered, and the Old Testament sacrifices that required the shedding of blood were only doing so as a way of prefiguring the fact that Christ would shed His blood. Christ wasn't completing the Old Testament sacrifices; the Old Testament sacrifices were acting as a "road sign" for the real thing.