Do I qualify for unemployment?

I gave my two week notice at my current job last week, this Friday was supposed to be my last day and then on Monday I was supposed to start my new job. This morning my new job emailed me to tell me my start date has been delayed by 2 weeks. Meaning I would have 2 weeks without income, which, unfortunately, I am not in a position to afford. My current job won't let me stay the extra two weeks. What are my options? Will I qualify for unemployment if I start a claim? Just for 2 weeks though. Or is it wrong to ask my new job if I can do some work for them (any work!) and start on my original hire date? Please help, I have 2 kids, a mortgage and a car payment among other bills, I can't afford to be out of work for those 2 weeks.

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