What a fun quest. I love gardening and would be happy to share what I know about real estate. Most lots in Austin are 0.16 acres... a few hidden gems are a quarter to a third acre. A larger lot would allow room for both a garden, chicken coop and possibly a shed too. My house is in South Austin - 78745. We still have homes in this price range, but they are getting more difficult to find. However, if you're okay with cosmetic issues and a smaller house, we could find something in your budget.
In addition to Austin, you could consider outlining areas that have more inventory in your price point. The older sections of Round Rock, Pfluggerville, Kyle and Bastrop have great lots for big gardens and chickens.
Do you have a location preference? Such as, must be withing X miles of work, school, hobbies? This also helps me guide folks to neighborhoods to meet their lifestyles. The great thing about Austin is most neighborhoods allow chickens.
Best Wishes in Your Search,