Were OT sacrifices sufficient for forgiveness? Sufficient how? Not in any ultimate sense. The OT sacrifices were most definitely nothing more than a picture of what was to come (see below for context). The reason I say this is because it is not defensible for God to forgive sin on the basis of animal sacrifices per se. God would be unjust to do so. In fact, that should be a massive tension for you when you read the OT. How, pray tell, is God justified in forgiving an adulterer and murderer like David on the basis of nothing more than some lamb getting its throat cut? He is not. David deserved death and hell; instead, all he had to do was butcher an animal and somehow he just blissfully says, “As for me, because I am innocent I will see your face; when I awake you will reveal yourself to me” (Ps 17:15, NET) Who cares if that was written before his greater sins: who does this guy think he is, and what kind of God would grant such felicity to such a man as one who committed, or would commit (would not God know what’s coming?) such outrages?. Something more is demanded if God is God: because if the God of the Bible is not just, he is not God.
Paul felt the tension. In fact, resolving that tension is the main burden of his argument in the first five chapters of his letter to the Roman Christians. Luther missed the boat when he saw these chapters as primarily an explanation of the justification of man before God by faith. These chapters are primarily an explanation of how it is that God was just in simply pardoning hell-deserving sinners like David, and how he is just in simply pardoning Christians on this side of the cross who are nothing more than hell deserving David-like sinners. This can be seen most readily in the climax of Paul’s argument in these chapters when he explains that Jesus being sacrificed to appease God’s wrath was to demonstrate: God’s conformity to the standard of himself (viz., his righteousness) {Rom 3:25, what follows through chapter five is further explanation of the dynamics and validity of this demonstration}. Why would God need to demonstrate to anyone his conformity to himself? For one, because it is totally illegitimate to simply “let go of,” “pass over,” or “wink at” all the sin of all those who experienced forgiveness before Jesus died, let alone just David. Jesus’ suffering the wrath of God against sin when he was nailed to the cross was, among other things, about providing proof that God was righteous in forgiving people like David before Jesus so suffered.
Without Jesus bearing the wrath of God against sin on the cross, all the bloody OT sacrifices amount to nothing more than animal cruelty.
It is a glorious truth that Jesus’ wrath-bearing death on the cross is also the proof that God conforms to himself (viz., is righteous) in forgiving hell deserving, David-like sinners at the present time. Jesus’ faithfulness = God is righteous in simply pardoning wicked, evil, sinful, hell-deserving men by grace through faith (main point of Romans 1-5).