AUSTIN, Texas—In 10 years of off-and-on homelessness, Rebecca Wallace has lived in the shadows of Texas’ capital city. Once, while living behind a shopping mall, she said a beating by a group of men put her in intensive care for weeks.

Now, Ms. Wallace sleeps on an inflatable mattress under a freeway in the city’s busy tourist district, along with a growing number of other homeless people. She moved to that area after Austin in July became the first major Texas city to allow public camping, in practice allowing people to reside...