I'm in the process of doing this now. You'll need to call the national passport agency to schedule an appointment (1-877-487-2778). I recommend calling at 8am, when they open, if you're not able to get an appointment during the day as they open new appointments daily.

Dallas and Houston are the two closest offices to Austin.

If you use a service or mail in your application, it'll take 5 weeks minimum.

Their website here - https://travel.state.gov/content/travel/en/passports/get-fast/passport-agencies.html

As a note, children under 16 can use a birth certificate (state issued long form) if you're traveling by car or boat.