I think Austin is fairly doable by bike (especially the central city). The city produces a downloadable bike map, which should give you an idea of how hard it would be to get from place to place. AustinCareShare.org is dead, but Austin Car 2 Go is starting up.

If you commute during the hotter months, and it's more than a couple miles, you'll definitely want to shower. As others have said, many of the high-speed roads have substantial shoulders, and Texas law explicitly allows for bicycle traffic on the shoulders. Motorists tend to be bicycle aware, too.