What is the next step when your FMLA is used up? If I quit my job can I file for unemployment?

I am on week 4 or 5 of FMLA and only allowed 12 weeks. My mom is in declining health. She was diagnosed with cancer in June of this year and was doing fine until a bladder infection knocked her off her feet in August and then another one three weeks ago has weakened her even more. She is on Hospice and a nurse does come 3 times a week or more if needed. A healthcare aide is coming daily beginning next week, but can only be here an hour at a time. She is basically bedridden and barely has the strength to use the bedside camode. My brother helps during the day as much as he can but works second shift and his wife fills in in the evenings and weekends when she can. My older brother and sister show up when it's convenient for them. So I am the primary because I live next door and my job is the most flexible. I feel very lucky to be able to see my mom through her last stages of life and wouldn't trade it for the world although it hurts to see her so weak and seemingly shrinking before my eyes everyday. I am grateful for everything she has done for me and my children and feel this is only just that I take care of her in her time of need no matter how long it may be. My dad is 87 years old and has to use a walker to get around and has the usual aches and pains that come along with his age. He is able to be a good companion to my mom but can only do minimal things to help. Our situation is not dire by any means but I would like to know my options so that I can continue to care for my mom without worrying about my finances.