As a fellow student, I know what you're going through. I was planning to take a CNA class this summer but it ended up not working out so I cannot take it until the fall when I will already be in nursing school. I am still going to try to get some part time work as a PCT because I know that the hospitals where we do our nursing clinicals usually have spots open.
Usually it will take a few weeks for people to respond to job postings. Especially if they are online in my experience. I don't think there would be anything wrong with applying for jobs now and just including in your cover letter of your resume that you will not be certified until that date. But, knowing the hospitals in my area, it will probably take that long for them to get back to you anyway and you would be all set.
I would list the time you spent with your ill family members as some experience in your cover letter, or if you don't do that then I would make sure you bring it up in your interview if you get one.
Based on all the problems I've been hearing with the new grads in my area, I'll be happy to take a job wherever I can get one after graduation, even though I would also like to work in a hospital. I think a lot of time though, people aren't willing to be flexible and go outside their comfort zone. They just apply to the same hospital over and over and don't try any other hospitals that might be out of their normal range. I know that there are a lot of job openings in the rural areas of my state, and if I need to commute for a couple of years to get my foot in the door, I will gladly do it. I just don't think a lot of people have that mindset or want to venture out.
Good luck with everything!!