This is a problem not just in Austin, but across the country that is not going to disappear any time soon as more and more Baby Boomers are retiring while the number of doctors that accept Medicare either stays the same or potentially decreases.
However, whether it be Austin or elsewhere, general guidance has been to be patient and realize the first few doctors you reach out to may not accept Medicare, but there certainly are doctors in the area that will. A few examples in Austin:
If you go to https://www.medicare.gov/physiciancompare/search.html, and type in Austin, TX, you'll see a notation for doctors that accept Medicare.
Besides the official government site, ADC is a massive clinic in the area that lists the doctors who accept Medicare: https://www.adclinic.com/patient-tools/insurance-2/insurances-accepted/medicare/doctors-who-accept-medicare/#.V-yBl5MrI6V