I don't know if this is possible in Austin, but Uber eats allows bicycle delivery (almost definitely available in Austin), and "on foot" delivery as well. Other delivery companies may have similar programs.

Instacart (in some locations) allows you to sign up just to do the shopping (then the drivers pick it up).

Taskrabbit will definitely have some things you can do without driving as well.

Just explore the options; something where you can set your own schedule would be really nice while you're job hunting and interviewing.