Am I religiously obligated to alert those who publish fabricated and weak Hadiths online, or is it permissible for me to remain silent due to not being a scholar, especially with the fear of negative consequences or the difficulty of distinguishing between mixed types of Hadiths, and does this obligation include monitoring the pages of relatives?

You are not obligated to monitor the pages of your relatives or others, nor to warn the publishers of unauthentic Ahadith, unless there is no one else doing that but you, as Al-Nawawi mentioned. Go easy on yourself and turn away from obsessive thoughts; for their consequences are dire. And among what helps you get rid of them are seeking Allah's help, sincere recourse to Him, and frequent supplication.