Emily - SW gave you some excellent advice about nursing training - the demand is huge and only going to grow as we boomers get old.
(And btw Emily, I don't know how your thread became a referendum on the welfare system, but I totally understand how it is necessary for a mom of two small children abandoned by her husband. Don't ever feel bad about taking that help. I also worry that you may be suffering from some depression - not a big surprise given the circumstances and having had a baby this year - but maybe you should discuss the possibility with your doctor? They might be able to help you with the mood swings.)
As for a job right now - lots of jobs never make it to those listings. Will the state pay for child care if you return to work? If so, type up a resume and go knocking on doors - visit every business in town and leave your resume with the manager. Someone just might be impressed enough with your drive to give you a job.
If child care won't be paid for - consider looking for a job in a preschool. Some of them will let the teachers bring their own kids for free or reduced rate. Or consider waitressing in the evenings? These are temporary fixes, of course.
You can do this. Try thinking outside the box a little, get creative - you're a bright girl.
