I'm sorry to hear about your situation. I know an amazing agent who works the Katy area. His name is Gilbert Nunez. His phone number is 281-689-1500. His office is in Cinco Ranch and he knows the area inside and out.
I'm very sorry to hear about your predicament. I've seen this happen to a few single mothers that I know recently and my heart always goes out to them. For the best and most successful method of attack, you do need to find a Realtor in that area that will help you search and locate another property fast. Also, I'd suggest having a basic idea of what your credit score is and explain most of your payment histories to your Realtor. That Realtor should then make sure they talk to the management company first about your predicament before you start plopping tons of applications down AND app fees.
Hope that helps.