Airbob covered it pretty well, but negated to mention Hidden Falls to the west. Its open Thurs-Sun (unless you have a membership) and you can ride there in most any weather conditions. I've ridden it numerous times during & after the rain. Its chunky rocky technical riding. But that also means that there's less mud and it drains well.
http://www.hiddenfallsadventurepark.comA good number of people in my riding crew also ride mountain bikes. Its another way to get your 2 wheel off road fix in, in a different manner.
Welcome to Austin. Hope you didn't drive the home values up in my neighborhood!
-CJB