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Do not include anything for a "rental agency" kind of place. You're going to need to rent directly from a person: either rent a room (search craigslist) or rent a place from a direct person. Don't include the previous tenant that is evicting you.
Here's the thing: eviction is a criminal procedure. It is on your record, and will be found in any background check. All rental agency places will require such a background check.

My overall advice remains the same as before: hope for the best, but prepare for the worst. Invest in a tent so that worst case, you and your pets have a home whilst you work on getting back on your feet.
As far as showers and shaving -- you can join many gyms for $1 / first month, and maybe $30 / mo thereafter. Otherwise, you can survive and be clean off "travel wipes" and public restrooms.
It's not ideal, but if you don't plan to handle failure, and it comes... it's going to hit like a freight train of bricks.
Hope this helps, best of luck to you!!!