Don't lose hope. I am a Realtor in the South East part of town and work with many investors who own rental properties. With poor credit, I can most likely get you into a home, but may have to put down deposit, first and last month.
Also, my wife is a leasing agent here and has 1-2 bedroom apartments that may fit your needs.
Keep in mind that Houston is huge and when you add to the city, areas like Clear Lake, Pearland, Katy and Spring (to name a few), you have a lot of options.
Feel free to contact me if I can be of any help! Thanks!
(1) Your budget is totally unrealistic ... $600 will likely get you a 500 sq. ft. studio apartment at best. You'd be fortunate to find something that would work for you at double that rent.
(2) Your poor credit will eliminate many options, irrespective of price. Most Realtors would decline to work with you, given the very small fee earned, as well as your low FICO score ... not to mention the fact that finding anything that would fit your budget would be looking for a needle in a haystack.
I suggest you try Craigslist or any of the local "green sheets." Landlords who work with Realtors usually insist on creditworthy tenants with excellent employment and rental histories. If your income falls within the Section 8 parameters, it's probably your best option. I wish you well.
Trend Setter Realty