What should I do, as I have become desperate, broken, and lost hope after being falsely accused of stealing my sister-in-law's gold?

Accusing people based solely on suspicion is forbidden. Allah says: (O YOU WHO BELIEVE! AVOID MUCH SUSPICION; INDEED SOME SUSPICION IS A SIN) [Al-Hujurat: 12]. You should not cut ties of kinship due to injustice; rather, you should pardon and forgive. Allah says: (AND THOSE WHO PARDON THE PEOPLE - AND ALLAH LOVES THE DOERS OF GOOD) [Aal-Imran: 134]. Therefore, repel evil with that which is better and meet ill-treatment with good deeds. If your not entering their home is to avoid accusation, there is no harm in that, provided you maintain ties with them and inquire about them. A true believer does not despair in times of hardship; rather, he perseveres, seeks refuge in Allah, and frequently seeks forgiveness. Many prophets were accused while they were innocent.