Is it obligatory to re-publish previously disseminated hadiths that have been clarified as weak or inauthentic, along with an explanation of their status, or is it sufficient to delete them automatically?

Sincere repentance is sufficient for you, along with the firm resolve not to disseminate anything until its authenticity is verified. You can ascertain the veracity of information by asking people of knowledge or by emulating trusted websites. You are not obligated concerning what has passed, as long as there is clear hardship in the matter.