I need to make an appointment in Austin Texas to replace my damaged MacBook pro battery

We are end users like you and cannot make appointment so your behalf.

First make certain Apple still provides battery service for your Macbook Pro sub-model:

Mac Repair & Service - Apple Support

using the service cost estimator at the bottom of the page.

Note that almost no Apple retail stores can replace the battery of a Retina Macbook Pro on site. The glue used to hold the battery to the upoer desk requires nasty solvents that cannot be used in a public or retail space without violating a raft of federal and state workplace safey regulations. Apple will likely have to send it to one of their centralized repair depots, adding a day at each end if you drop it off at the Store.

I use the mail-in/ship is option. WhenI set up a repair it gave me the option to drop my Apple device at a neighborhood UPS Store five minute from my home. I set it up on the Apple site and, when I got to the UPS Store, they had all the details from Apple and even packed and packaged the device at no extra charge. That included return shipping to my doorstep.

NOTE: If the battery has swollen and caused other damage, the cost can be far higher and even irreparable. A swelling battery can damage many more things than just the bottom plate.

Everything starts with Official Apple Support.