How can one overcome a state of despair, unhappiness, and feeling unloved by others, while possibly falling short in one's duties towards Allah or oneself?

A good and happy life is achieved through faith and righteous deeds. Allah loves the repentance of His servant. To deal with the feeling of not being loved by people, treat them kindly. Do not let your concern be the love of people, but rather the love and pleasure of Allah. For pleasing people is an unattainable goal. If you please Allah, and He is pleased with you, He will make people pleased with you. And if you displease Him, people will be displeased with you. It is recommended to read the book (Don't Be Sad).