By mental faculties I mean when people get old they often lose some of their brain power. Even if it is not as obviously severe as Alzheimer's, they can't remember things as well, or they can't think things out as well. Their ability to take care of themselves go down, and often they rely on others to help them.. Sometimes they become more fearful of burglars and stuff like that too.
So I guess what I am asking is did she get used to you doing stuff for her, or rely on you for things, and therefore need you? Often old people do not like to admit this and will make it sound like something else (like needing to look after you.)
I asked about the roommate because if she needs a roommate to do things for her or for financial reasons but does not get along with her, that might be a reason to want to live with you.
She may not be as close to or used to your cousin and prefer you.
These are just other possible reasons. I am not saying that is what is going on. It may just be that she worries about you and it would fix her anxiety if she could constantly check up on you. It does not mean you have to put up with it, though. You may just have to repeatedly tell her you can take care of yourself, you've done it before and you want to be given space. Sometimes people need to hear things multiple times, and it has to be very direct.