Well - not exactly...
It's incredibly important to remember that there is NO condemnation for those who are in Christ Jesus. Paul, in Romans 8:1 makes it clear to us that:
... there is no condemnation for those who belong to Christ Jesus.
You are righteous before the living God. Jesus, came and suffered our punishment for us. We have already passed from death to life and do not face condemnation from the Father.
Even though many of us know this truth, we still continue to struggle with out past sin. In Philippians 3:12-14 Paul says:
I don’t mean to say that I have already achieved these things or that I have already reached perfection. But I press on to possess that perfection for which Christ Jesus first possessed me. No, dear brothers and sisters, I have not achieved it, but I focus on this one thing: Forgetting the past and looking forward to what lies ahead, I press on to reach the end of the race and receive the heavenly prize for which God, through Christ Jesus, is calling us.
Paul in this passage is stating that he forgets the awful memories of persecuting Christians and rebellion against God. He looks forward; he doesn't dwell or wallow in the past. But he also says in Ephesians 2:11-12:
Don’t forget that you Gentiles used to be outsiders. You were called “uncircumcised heathens” by the Jews, who were proud of their circumcision, even though it affected only their bodies and not their hearts. In those days you were living apart from Christ. You were excluded from citizenship among the people of Israel, and you did not know the covenant promises God had made to them. You lived in this world without God and without hope.
These are two different messages that look contradictory of each other at first, but they actually go hand in hand. Paul is saying that we should forget our sins and press on, remembering them only for a deeper understanding of God's grace and mercy. Paul calls us not to be paralyzed by our sin, but to praise God and come back to Him in thanks for his mercy and forgiveness.
It's imperative to recognise that wallowing in your own sin is also sin - by doing this, you are idolising yourself and saying that Jesus' forgiveness was not enough for you. What could possibly be more offensive in sight of the Heavenly Father, the Creator of all things, the almighty and righteous God. I myself, am terribly guilty of this and seek repentance for this sin often.
So should we forgive ourselves?
Why would you forgive yourself if you trespass against someone else? The Bible makes it clear that we should seek forgiveness from those that we have wronged and from our Heavenly Father.* Who made us the judge and executioner of our own sin? As Paul said, there is no condemnation for those who belong to Christ.
Paul in 2 Corinthians 7:8-10 talks about how the Corinthians repented and turned away from their wicked ways. Paul had previously called for their repentance, and he endears them for seeking reconciliation with the Father.
I am not sorry that I sent that severe letter to you, though I was sorry at first, for I know it was painful to you for a little while. Now I am glad I sent it, not because it hurt you, but because the pain caused you to repent and change your ways. It was the kind of sorrow God wants his people to have, so you were not harmed by us in any way. For the kind of sorrow God wants us to experience leads us away from sin and results in salvation. There’s no regret for that kind of sorrow. But worldly sorrow, which lacks repentance, results in spiritual death.
Paul is clearly stating that you should have no regret for the God inspired sorrow you may feel about living a sinful life prior to salvation, but worldly sorrow tells of a lack of repentance and results in spiritual death.
He continues on to say that the Corinthians godly grief that they had suffered through produced repentance that lead to salvation without regret. Now they too can continue to look forward and toward the future that Christ has laid out for them.
Just see what this godly sorrow produced in you! Such earnestness, such concern to clear yourselves, such indignation, such alarm, such longing to see me, such zeal, and such a readiness to punish wrong. You showed that you have done everything necessary to make things right.
Taking all this into consideration, the Bible tells us that we do not necessarily forget our sins, but that we should strive to forget our sins and press on, remembering them only for a deeper understanding of God's grace and mercy, and to humble ourselves and step down from our seat of condemnation over our own life.
One more thing to consider.
We are sons and daughters of the most high, we co-heir with Christ and sit at the right hand of God. It brings no glory to the creator of the universe if we sit and do nothing with our lives because we believe ourselves not worthy of existence. You have been SET FREE THROUGH JESUS' SACRIFICE! It's time to start living freely from sin and condemnation! God did not design us to suffer alone, and if you still continue to struggle with your past sin, I endear you to find a mentor or a friend that can hold you accountable.
* Only if the person you have sinned against knows you have sinned against them. For it would not make sense to ask for forgiveness from someone if you committed adultery with said person in your mind for that may cause unnecessary divides and relational issues producing additional sin and harming the person.