Other available options:

Have you posted on For the Love of Animals in SETX? There are a lot of local rescuers on this Facebook page who may be able to assist. https://www.facebook.com/groups/1684155595197936/

Have you submitted a surrender form to the Humane Society of Southeast Texas yet? If not, you can do so at https://www.humanesocietyofsoutheasttexas.org/surrender/
If you or someone you know is able to foster, they may be able to assist even if they are full. Rescues are sometimes able to help quicker if you are able to foster while they help place the animals.
Please be sure to include your ability to foster on the application. If you are unable to foster, they may still be able to assist you. However, you will still need to submit the owner surrender form online.
Then they will follow up with you to let you know whether they can assist.

Below are some additional rescue groups to contact. I may be able to suggest someone who could help transport if you are unable to transport.

Local Rescues
PAWS Beaumont – https://www.facebook.com/PAWSBeaumontTX/
Guardians for the Animals – https://www.facebook.com/GuardiansfortheAnimals/
Pups in Peril, https://www.facebook.com/PupsInPeril/
Adopt Me Meow, https://adoptmemeow.org/
Meezer’s Mission Cat Rescue, https://www.facebook.com/MeezersMissionCatRescue/
Sweet Pups, https://www.facebook.com/sweetpupsdogs/

Other Areas
Austin Pets Alive (512) 961-6519
Austin Humane Society (512) 646-7387
Houston Humane Society (713) 433-6421
CAP Animal Shelter Houston (281) 497-0591