This is very good. You do not want to either leave her unsupported nor mess up your time in university. I am not a grandfather yet but am old enough to be, and would not want to mess up the education of a child or grandchild due to the need for someone to care for me.
Students often under-appreciate the quality of their in-state public universities, and over-estimate the importance of small differences in rankings (which can change over time, or from one magazine to the next). You might want to start a different chance-me thread if you want suggestions for schools to apply to. There are of course a huge number of very good universities in the US, and more elsewhere.
Best wishes.