There is an animal trustees in Austin however they do not see sick animals. They do have some special programs that might see your cat though. You will have to call and ask. The number is ###-###-####. Your other option is to do something like Care Credit. I did this a few years ago when my dog was very sick. Basically it is a low interest rate credit card. Sometimes they even let you do a 0% interest. Care Credit then pays your vet bill and then you can pay back on a monthly basis. Or another option is to call around and find a vet that will let you do a payment plan.
I hope you find treatment for your beloved family pet. My dog ended up passing away so I know how hard it is to lose a pet.