Unfortunately it looks as if it would be mathematically impossible to qualify based on the info provided. Good luck in finding steady work in the near future.
Unfortunately, do to the past due payments on your mortgage you will not be able to refinance even with 20% down. I would report that agency to the BBB. I'm sorry for the poor advice that someone has provided you with.
It's a no brainier, USDA. USDA loans take the same amount of work as an FHA maybe a little bit more. A lender that's looking out for you will do what takes to make that happen. I hope it works out for you.