Inky i am sorry to hear of your situation,it is never easy to not have a roof over our heads!
I would say Target is not a good place for you,you seem to have anxiety just remembering how they treated you b4 ?BUT if you can just hang out long enough to find another gig,then go4it.
Navy job,does the move pay4your kids 2 relocate too?You may consider quality of life even if the pay's lower?Maybe you can also take advantage of their training programs.Then you can get a better pay rate.Guess a job with a future, education upgrades is a better option?
If you are homeless,are there any state funds for you to repair your car somehow?Sometimes if you are disabled, there's funding4medical transportation.Do you know anyone who is a mechanic,pay4the parts?Labor is what kills you4car repairs

I loved Chicago when i visited,but i know it's very pricey.