use Doobert.com!
Hopefully, the shelter will allow you to place a hold on the cat, submit any required authorizations and pay any fees electronically. Then set up transport from shelter to shelter. If the shelter requires in-person authorizations, you should be able to use Doobert.com to partner with a local group, too...Austin Pets Alive! should be a great resource in that area. TX to AZ shouldn't be too difficult to get volunteers for legs. I would offer myself but I'm in GA.
GOOD LUCK and thank you for getting kitty back to safety! ^.~.^
#TransfersandTransport