Is not searching for a paper on which sins are written—due to its existence in a forgotten place among much luggage—considered a public disclosure of sin?

Writing down sins as a means of self-accountability and reminding oneself of the sin's gravity is permissible, and it does not fall under the category of openly committing sins. This is because the state of the writer, with that good intention, differs from the state of the one who openly commits sins, who reveals his transgression and uncovers what Allah has concealed for him without necessity or need.