On the local level, most cities have rental offices that assist residents with their apartment search. When you go in, be up front not embarrassed about your financial concerns and needs. That way, you'll be placed on a call list to get notification whenever a low income or low bond apartment opens up. Most apartment complexes reserve a few units for this and you'll want to have a shot at renting one when it becomes available. There's no sense wasting your time being called on rentals that are out of your range.
How To Qualify for Low Income Housing Waiting Lists
On the federal level, the Housing and Urban Development (HUD) website (https://www.hud.gov) has a handy rental locator search engine. Simply put in your zip code or city of choice and you will get a complete listing of rentals in your area that are considered low income or affordable housing rates. You must contact each one to see if they are taking new people on their waiting list and then you will need to go in person to get on the list. Move quickly because the lists can be years long. Don't get discouraged. The same time will pass; it's just a matter of being patient and trying to have decent living arrangements in the mean time.
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Once you choose the apartment complex that fits your needs and you get initially on the waiting or call list, go in person often to see if you've moved up on the waiting list. It makes a good impression and rumor has it that it can help you move up the list although that's unofficial; just what the word is on the street.
Once you get that all important call that affordable housing has opened up for you, be prepared to move in because you only have a week or so to come up with the money or they move on to the next person on the list. The usual requirement is first and last months rent and sometimes a deposit is added in. So watch your savings and be prepared should your turn come.