He should get unemployment but he's neglected to complete the paperwork correctly. He said yesterday that he finally finished it and so I think the payments should start in a week or two (assuming he truly did finish the paperwork). Other than that, no income unless someone here gets a job.
What happens between now and then? How will you guys eat? Sometimes it can take up to three or four weeks for unemployment checks start coming in (assuming he truly did finish the paperwork). What if he didn't finish it?
Do you have family around? Could they watch the kids while you work? Could you and the kids move in with family while you look for work? Something has to give. You have children and the most important thing is their welfare.