Credit is only one of the landlord's considerations in making a rental. They consider criminal backgrounds, having a steady job, etc.
I'm a Dallas bankruptcy attorney and out of the 1,500 cases I've handled, I don't think I've ever had a client turned away from renting a place because of their bankruptcy.
If someone is turned down for an apartment based on their credit, it's because of a low overall FICO or Beacon score. Your bankruptcy probably raised your credit score, not lowered it. If your credit held you back from an apartment now, it would be because of other issues on the credit report rather than just the simple fact that you filed a bankruptcy case at one point.
An apartment complex is not the only way to go, of course. There are around 2,500 houses on the foreclosure list every month. Even more if you count the surrounding areas and not just Dallas County itself. After each monthly foreclosure sale, there are tons of new properties available for rent. That's in addition to the rental homes that already existed.
Some of these rent houses are managed by a professional property management company, where there will be a formal application process and a look at the credit report. The mere fact of the bankruptcy should not influence the decision although the overall credit score will.
Other of these rent houses are owned by individuals who know the amount of rent they'd like to receive but do not have a formal application process. They won't bother to even pull the credit report. In fact, individual landlords don't have any way to legally get their hands on one (Experian, Equifax and TransUnion don't just hand out your credit report to individuals who call them and ask for it. That's illegal under the federal Fair Credit Reporting Act). Even if you sign an Authorization allowing them to look at your credit report, they still don't have a way to legally pull it. There's a whole licensing procedure for that and individuals don't qualify.
The Dallas housing market is very diverse. There are rentals available for a wide range of prices, for all types of credit, in a broad range of locations. You're very smart for trying to get potential issues dealt with ahead of your big move, but I wouldn't worry too much about this part of it.