All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
If the matter is as you mentioned in the question that your husband is accusing you of establishing an illicit relationship with other men and he has no evidence about this, then he has indeed committed a major sin and he should be advised and reminded of Allaah so that he would stop propagating such statements. One should threaten him about taking the matter to the authorities if he does not stop doing so.
We advise you to cover yourself (by Hijab) and abide by the religious obligations when dealing with non-Mahram men, and avoid places of suspicion so that people would not think that your husband’s rumours about you are correct.
If you hate your husband, then your parents have no right to oblige you to stay with him since you are harmed by staying with him. However, if you can be patient with your husband and continue to advise him with the hope that he will become righteous, then this may be better than asking for divorce for the sake of your children and in order to be kind and dutiful to your parents.
Nonetheless, if the marital life with him is impossible, then you may ask for divorce even in return for compensation. If he does not accept, you may take the matter to a Muslim judge so that he will oblige him to divorce you or to grant you Khul’.
Finally, it should be noted that it is not permissible for you to go out of your husband’s home except with his permission unless going out is for repelling his harm and the like, in which case you may go out to the house of your guardian (father and the like). As regards running away to any other place, then this is not permissible.
Allaah Knows best.