Once a Christian asks for forgiveness, is the sin forgiven and forgotten by our Lord?

Yes. That is not to say that the results of that sin do not remain in this life. For instance, the murdered person is not brought to life and the sexually loose person may still deal with the diseases they caught. However, in God's eyes, they are gone.

One illustration is found in the Samaritan woman. Jesus told her, "For thou hast had five husbands; and he whom thou now hast is not thy husband: in that saidst thou truly" (John 4:18For thou hast had five husbands; and he whom thou now hast is not thy husband: in that saidst thou truly.
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She had had five husbands. However, Jesus did not tell her that she had five living husbands. Those husbands were now past tense. Now, she was living with a man who was not her husband. That was her present sin.

The concept of "living in adultery" has created a false burden of continuous sin for people who have wrongly married a second time. If their marriage is adultery, then it is a sin that needs the forgiveness of God. However, if the marriage is a continual state of adultery, then the only way for them to get right with God is to divorce their last spouse no matter how many children they have or what else may happen. But God never commands us to do this. God always starts with us where we are and allows each person a full opportunity to be right with Him. He does not create scenarios where it is impossible to be out of sin. This situation has been created by men.