Is touching the utensil that I swore by divorce not to use for cleaning, after finishing the cleaning, whether the touching was forgetfully or intentionally, considered a breach of the oath, knowing that the oath was intended only not to use it for cleaning and not not to touch it or use it for life?

The divorce you mentioned does not take effect because you did not intend it with your oath. The occurrence of an action or statement in a manner different from what the swearer intended does not constitute a breach of the oath, nor does it result in divorce. You should know that it is not permissible to swear by anyone other than Allah or His names and attributes, and swearing by divorce is blameworthy. Allah Almighty has commanded dutifulness to parents and treating them with kindness. Showing boredom, annoyance, and expressing 'uff' (a sigh of impatience) towards them is a form of undutifulness. Therefore, seek Allah's forgiveness for what you did to your mother, ask for her apology, kiss her head, and do not repeat such an action.