Sorry, I am not able to see how your current home life situation would affect qualifying for low-income housing. Neither would your mental illness status.

Are you in physical danger? Because they may have alternate temporary means of sheltering you.

I have applied for low-income housing myself. My then current living situation had nothing to do with my application. I was interviewed entirely on the basis of my ability to earn an income and my ability to pay my then current rent and living expenses. I was placed in a queue according to my need accordingly. I never did get selected. My income was considered low enough but the fact I had a strong support network kept me down the list.