Brandon...
Experience talking here, the jobs would contact you via your cell. You will have your cell with you. These jobs are with people, not robots. When they call and you are caring for family, tell them that. Say "thank you very much, I would enjoy starting right away. Howerver I am in Redneck South as my very close grandmother passed and I will be back in one (or two) week and can start at that time."
Believe me, people pass on in this world and many have also experienced it. They have compassion, it is understandable that you are handling things as you should and wish to have a job as well. At one point in my career, I made the bosses aware of things happening in my family. One day I left work and didn't return for nearly a month. I did keep them in the loop as to when I believed I would be back etc. These people said to me, "don't worry about work, it will be here when you have taken care of things.". Heck, I was also mid semester and had made a mention to the professors, they also gave me a lot of leniency in assignments etc.
When they call, be upfront and they should have no problems with a week or two. Afterall, many people give notice to a current employer before starting with the next. Two weeks shouldn't be a make or break on a job. If it is...you don't want that job. See your SuperCuts experience in the past. Not the kind of boss anyone would want.