Is it permissible to speak about a husband's injustice and its related details, even if it is in response to false accusations? And does sin result from that?

It is permissible for the oppressed to complain to the ruler, the judge, or anyone in authority or capable of redressing the injustice against them by the oppressor. This is one of the situations in which backbiting (Gheebah) is permissible. It is also permissible to describe the oppressor with what is true about them, by way of retribution, without lying or aggression, though refraining from this is better. However, mere divorce is not an injustice if the rights of Allah are fulfilled therein. When afflicted by a calamity, one should say: "Indeed we belong to Allah, and indeed to Him we will return. O Allah, reward me in my calamity and replace it with something better."