Is the wife obligated to visit and serve those who spread false rumors about her, knowing that the husband considers this normal and she fears it will destroy her psychologically?

If what you mentioned about your husband and his family abusing you and spreading rumors about you is true, then they are in the wrong. Forgiveness is a great act of devotion, and efforts should be made towards reconciliation, especially between relatives and in-laws. Trust and affection must be present between you and your husband to maintain family stability. You are not religiously obligated to serve your husband's family, especially if you fear harm. And if your husband falls short in his duties towards you, it is not permissible for you to fall short in your duties towards him.