Is it obligatory to search household waste to remove papers containing the name "Abdullah" or the name of Allah (Jalalah) to avoid their desecration, especially when the family members do not care about that?

Thanks to the questioner for her keenness to glorify the names of Allah Almighty and protect them from trivialization. The names of Allah Almighty must be glorified within the objectives of Sharia, including the removal of hardship and difficulty, for Allah Almighty says: "Allah does not intend to make any difficulty for you" and "And He has not placed upon you in religion any difficulty," and for the Prophet, peace and blessings be upon him, said: "Indeed, religion is ease." What you mentioned is a widespread affliction from which it is difficult to guard against. Therefore, papers on which the name of Allah or anything else glorified by Sharia is written should be preserved in a way that involves no hardship, and you are not required to search for these papers or inquire about them.