Is the gift-giving to an offending other person a way to prove to yourself that you, indeed, have forgiven?

The gift-giving in its essence is not for the forgiver, but instead is for the one forgiven.  Forgiveness as a moral virtue is concerned with goodness and that goodness flows out of the forgiver to the forgiven.  While the gift-giving can be a sign to you that you have forgiven, that is not its primary function.  The primary function is to do good to the other as a moral act in and of itself.

Learn more about what forgiveness is and is not at What Is Forgiveness?