If you are renting from a large concern, your credit score may be an issue. But there are many "small time" where owners couldn't be bothered to get involved with this issue. So yes, you should be able to find something.
I sold homes for several years and I also was a mortgage professional in Texas for some time. I would say the your references on your rental history would be crucial. I would also expect a bit larger deposit due to the risk. There are ways to get your credit score up as well. I can help youi raise your score or teach you how to do it yourself. Feel free to email me anytime if you want some honest info. email@example.com. Have A Blessed Day!