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, sallallaahu ‘alayhi wa sallam, is His slave and messenger.
If the case is as highlighted in the question and you are still married to your husband, you are obliged to obey him as long as it does not involve disobedience to Allaah. Hence, you are not allowed to work without the consent of your husband. However, if he does not fulfill his duty in providing for you, it is permissible for you to work without his permission, according to Muslim jurists. The Shaafi‘i scholar Zakariyya Al-Ansaari wrote (in Asna Al-Mataalib), “When a wife leaves her house without her husband's consent, she is considered Naashiz (recalcitrant). However, if she leaves the house in order to refer her case to the judge and to complain that her husband does not support her .... or in order to work because he does not provide for her (i.e. it is not considered recalcitrance on her part in these two cases).”
In fact, the reason you have mentioned that your father and brother are having financial straits is not a valid excuse for you to work without the consent of your husband. The same applies to your grandfather’s illness. Hence, there is no harm in seeking the permission of your husband and trying to convince him to allow you to work.
It should be noted that Khul’ does not take effect by merely referring the case to the court unless the judge has issued it. Also, a valid Khul’ does not require (the ruling of) the judge; it can take place by the mere mutual consent of the spouses. However, the formula of Khul’ is a condition for its validity. The Hanbali scholar Al-Buhooti wrote, “The formula of Khul’ is essential in all forms of Khul’. A husband may say, ‘I give you Khul’’ … ‘I hereby dissolve this marriage with you for such-and-such compensation' ...’, or 'I annul the marriage with you for such-and-such compensation', and the wife answers, ‘I accept,’ for example, and thus the Khul’ takes effect. The same applies when she says to him, ‘Give me Khul’, ‘Divorce me,’ or the like and he responds saying, ‘I give you Khul’,’ or the like. These are explicit formulas of Khul’. Implicit formulas of Khul’, like ‘I relieve you’, or ‘I am separated from you’, may be used as well. Also, a third party may request the husband saying, ‘Give her Khul’,’ or ‘Divorce her,’ in return for one thousand (Dirhams) or the like and the latter responds to the request.” [Kash-shaaf Al-Qinaa‘]
For further information on the laws of Khul’, please refer to Fatwa 89039.
If Khul’ takes effect, it is considered an irrevocable divorce. In this case, the divorced wife is not entitled to financial support or accommodation unless she is pregnant, according to the most evident opinion of the Muslim jurists. If the divorced wife needs to work during her ‘Iddah, she may do so on condition that she spends the night in her house.
Allaah Knows best.