My own single mother is being avoided by me because she cares for my disabled sister.
Why would you avoid your mom for taking care of your sister?
I don't like that she cares for my sister more than me.
It's awful to avoid your mom for taking care of your special needs sister, who can't take care of yourself. Taking care of someone is a difficult job, and you should be supportive of your mom for doing such demanding work.
She cares for my sister more than... she cares for me.
Exactly, she cares for both of you. She just needs to care for your disabled sister more because of her disability, but that doesn't mean she doesn't love you both equally.