You do not qualify for unemployment.
Unemployment is for people actively looking for work and willing/able to accept work. You aren't. You say you are responsible for your grandma 24/7, so you aren't able to accept work.
Additionally, to receive unemployment you have to have worked two quarters out of the base period. Unless you, I'm missing something in your note, this is a first job and you worked less than a month.
Furthermore, unemployment is for people who lose their job through no fault of their own. You had a job available but you resigned from it.
If you need to work, your family needs to find other solutions to care for grandma or how to support you while you aren't working.