Each state has its own rules and procedures. The link above takes you to a clickable map that will help you find the right department to contact in your state.
And take a deep breath. If you think you're the first, or last, post-academic to seek government assistance, you're wrong. There's absolutely no shame in putting food on your table. There's absolutely no shame in admitting you need help. But it can feel humiliating to consider the disparity between the fantasies that drove us to pursue academia and the reality of our post-academic existence. I certainly didn't dream of living in a small apartment with a filthy carpet filling out paperwork for dependent care assistance when I sipped wine and talked theory in the wood-paneled dining room of a professor's bungalow at the first departmental party I attended as a graduate student.
The new reality can suck, but don't let that deter you from seeking help.
Post-academic in NYC writes about this in "The Crushing Shame of Applying for Unemployment:"
When you call the number, a person who oozes resignation and cold efficiency asks, “you had a teaching job last year, so why did you quit?” You are expected to have a really good reason for why you quit your adjunct gig that didn’t pay well to take a part-time gig that paid a little more (which has since kind of dried up). It is hard to explain this because you are talking to a person who probably thinks “college teaching” sounds like the best thing ever. You can tell the person on the other end of the phone is judging you. She thinks you are an idiot for giving up a perfectly good “college teacher” job, even if it was part-time for not a lot of money. She thinks you’d rather suck on the public teat than work for a living. You really want to launch into a speech explaining about how the neoliberal economic forces destroying the economy also ensure that most college teachers are low-paid adjuncts who live in caves and suck just enough water to survive off of damp surfaces. You also want to explain that surviving grad school and writing a dissertation means you are many things, but lazy isn't one of them.
Jessica Burke, an adjunct, also writes about her experience with an unemployment hearing here. Burke writes about a fairly intimidating and humiliating bureaucratic experience trying to get unemployment, but New Faculty Majority offered her a lot of support. New Faculty Majority has a lot of fantastic information for adjuncts applying for unemployment that you should definitely check out, including a PDF guide for contingent faculty who are seeking benefits. They are fierce advocates for the labor rights of adjunct and contingent faculty, and unemployment benefits are a big part of that. Be sure to let them know if and when you file for unemployment.
The bottom line is: you deserve support.