Is it obligatory for someone who has narrated a prophetic hadith without verifying its authenticity to do anything else after repenting and resolving to verify the authenticity of what they narrate in the future?

It appears from the questioner's inquiries that there are obsessive thoughts, doubts, and delusions. The cure for these lies in neglecting them and paying them no mind, for these are among the deeds of Satan by which he corrupts a Muslim's religion and worship. It is good to investigate what a person speaks about. A student of knowledge should continue seeking religious knowledge and sitting with scholars to gain insight and understanding into what they speak about. Allah Almighty says: "And do not pursue that of which you have no knowledge. Indeed, the hearing, the sight, and the heart - all of them will be questioned." And Allah Almighty said to His Prophet: "Say, 'This is my way: I invite to Allah with insight, I and those who follow me. And exalted is Allah; and I am not of those who associate others with Him.'" Insight here means clear explanation, undeniable proof, and certainty, which can only be achieved through knowledge.