There are two REI stores in Austin, including one down town. I have shipped my bike to the Missoula REI three times and had them assemble it (for a fee, of course) so it was ready and waiting for me when I arrived. They were happy to hold my box and duffel bag that I flew my gear out in. At the end of my tours, I rolled the bike back in, gave them my Bikeflitghts shipping labels and relaxed.
Give one of the local stores in Austin a call and see if they offer the same service. I would do so earlier rather than later so you can get on the schedule. The bike departments can get very busy certain times of the year, especially after big sales. People order bikes, the get shipped and then need to be assembled.
One note about Bikeflights: If you are going to be on the road for more than a short time and need return shipping you will need to get computer and printer access during or just after you finish. That's because Bikeflights won't email you your return shipping until 10 days before your ship date. That can be a PITA if you are on a tight schedule and are in rural areas without reliable Internet access. A couple of times I had to make a bee line to the Missoula public library at the end of tours to get my labels. Had to do the same in Rapid City, SD at the end of a tour.