Having been homeless before, would it help to look at this from a different angle?
Can you afford a working beater car before you have to be out?
Lucky for me, when I was homeless I still owned a car. As long as you own a car you can be warm, dry, and secure. And you can have your dogs with you. Check into shelters, friend's houses to find a regular place to shower and things so you're presentable for work.
Just remember that it's temporary. The situation you're in is temporary. Do whatever you can to keep your job. Your dogs won't mind living in a car with you. The weather's still cool enough that they'll be okay, or maybe you can find a friend who won't mind watching them in the day if you're worried.
It'll give you more time to find a good place to rent.
You could get your boss to write a letter to prospective landlords, or when you get paid go to the library and make photocopies of the checks to show them. The combination of the two should be enough for most landlords.
Good luck. ((hugs))