Help with injured dumped black cat

Hi. About 3 weeks ago my neighbor saw a car dump a black cat in our neighborhood. Not sure if the cat sustained injury when they were throwing her or if kitty was dumped because of her injury – kitty has a limp on her right back leg. No collar, not sure if boy or girl. Very chatty and loving. The kitty has been resting in my backyard for days, and I have been feeding her. She's friendly and allows me to pet her but overall scared. I have two cats who are extremely territorial or I'd have taken this sweet kitty in a heartbeat. She constantly wants to come inside while my cats hiss from the window. She rubs all over me and purrs loud when I go outside to feed her. Today I noticed her limp is getting progressively worse and other established community cats are picking on her. With the limp she's not able to even walk fast let alone run or fight. She's not going to make it out there on her own for too long.. Other than the limp she seems to be a healthy cat. Please kindly suggest injured cat rescue groups who will help with giving her a chance. She's a beautiful cat and very docile. It's a shame such a sweet kitty got dumped.. people are extra mean to black cats. I have called our local animal shelter and they said no and told me to leave her alone as a community cat. But she's a dumped cat with a limp.. not a community cat, she doesn't know this community or this community's other cats. I'm afraid we're running out of time to help her. I'd appreciate any leads on rescues who would give her a second chance in life. I live in the Sugar Land, TX area. Thank you.