Having been a landlord in the past I can say that depends. It could depend on the rental demand in your area, how nice a place you will be renting, how long the rental has been on the market, how much you are willing to sooth the nervous landlords nerves, your overall financial situation, your job, your demeanor, and the landlady's mood.! Seriously!
I am guessing you aren't feeling very strong and confident right now, it doesn't help to tell you how many other people are in the same situation either. The good new is, it will probably get better. It will probably be a huge load off your heavy shoulders to be out from under all that pressure and to be able to look to the future to move on. I'm sure you've heard you should be saving as much as possible right now. I can't stress this enough because of my following strategy.
Some places could very well turn you down flat, if so, take note and move on (check back with them in the unit is still on the market another 2 weeks). Smaller landlords will have more flexibility to make their decisions out of the box and they could be your best bet. I have rented to people I felt sorry for. This always made me regret my decision, yes more than once or twice! So I don't do that anymore, I need to pay my own bills and keep the utilities on.
I recommend that you be honest about your situation (which sounds like my previous mistakes) but the difference is you have enough of a deposit to cover the risk for the landlord. In the Seattle area it takes 3 months to evict a tenant IF the landlord does all the right paperwork on all the right days etc.(that means 3 months lost rent plus over 1K in legal costs).
I had a young lady approach me, said she had made mistakes, but was on a good track again.I believed her. She was tidy in appearance, was polite and respectful, I met her boyfriend (same adjectives) so I made a deal if she paid 3 months rent up front I would take the risk. I got 3 months notice before she moved out, and a great tenant who left the place in good condition. She got a strongly positive reference in return when she moved to a larger place. It made me feel good she appreciated me taking a chance on her and that I could help her on her way.
Sorry for the long response, just wanted to let you know the landlord's perspective. I recommend looking for the landlord that only have to abide by their own rules and intuition. My advice is to clean up your debts as much as possible in the meantime and still have a cushion of cash for deposits and background check fees. Best of luck to you!