My family member is mistreating her cat and I can’t find him a new home or shelter to go to.

I moved in with a family member and she has a cat and dog. When I’d come visit before I moved in the cat would be free roaming and get along just fine with her dog. She got into a depression and stopped cleaning her house and wouldn’t clean his litter and I’d come here and clean up and all, over time she started confining him to one room only and not letting him out because he began to pee and poo on her things and break things and knock them down and she couldn’t keep up with his mess and over time he started throwing up too and she can’t afford to take him to a vet anymore either as her dog was recently diagnosed with lymphoma and she is paying for her chemotherapy while in between jobs. I can’t afford to take the cat in as my own as I live with her and he stays in my room and i make about minimum wage. I’ve posted him on websites for our area and no one will take him I’ve posted him on facebook and payed for him to be on adoption websites but once I tell them he can only eat a specific soft food without throwing up they all bail. I tried reaching out to adoption shelters to surrender him but I’ve got rejected every time. Maybe I’m not wording things right? I don’t know what to say when applying as every shelter is very full taking in cats from the streets I feel so guilty for her poor cat and don’t know what I can do now. She won’t take any steps herself to take him anywhere and just tells me he needs a new home like he’s mine to figure out but I’m out of ideas.