Will I be sinful for throwing away bags with the name "Al-Aziz" on them in the trash, and must I search for them, knowing how difficult that would be?

There is no obligation to search through papers and garbage cans to ensure that God's name is not present therein, as this would lead to hardship and difficulty, which are negated in Islamic law. Your self-imposed strictness in this matter and other forms of blameworthy waswasa (whispers/obsessive thoughts), and the fear of falling into disbelief due to something you have not done, are also a form of waswasa. We advise you not to open this door for yourself, for whoever opens it falls prey to Satan, who complicates and obscures matters for them. The Prophet, peace and blessings be upon him, said: "The mutanatti'un (those who are excessive/rigorous/go beyond limits) are ruined," three times, meaning those who delve too deeply, are extreme, and exceed boundaries in their words and actions.