Maybe. I did something similar while living in San Diego, CA and looking for work in several states in the Midwest. However, I put in my resume summary (plus on Dice & Monster job boards) that I was moving to <city & state>, when I was moving, what areas (suburbs/cities/states) I was interested in working in, and when I would be available for interviewing. This seemed to generate a better response from recruiters in the states & cities I had listed, although some recruiters didn't bother reading my summary. Also, once I moved to the Twin Cities, I received more recruiter emails and more interviews. But it does take time so be patient.
Don't just contact specific companies. Also contact recruiters, temp agencies, and use the local Texas Workforce website to find jobs in Austin. Every state has a Workforce website to help the unemployed find work. Networking is the way to land many jobs in this economy - do you already know people in Austin that you can network with (share your resume, talk about jobs, etc)?
Keep in mind - companies all over the US are receiving hundreds of job applications, resumes, etc. Just keep trying - it took me over a year to find a job and I had to move out of CA to do it! Good luck!