The new track down near the old airport is called Driveway Austin. It's a really nice facility - they've even replicated several famous turns including "the corkscrew" from Laguna Seca and Festival Turn at Monaco.
http://www.drivewayaustin.com/
They're big on instruction, so it wouldn't be turning us loose on track but something more controlled I'm sure.
There actually was a track named Austin Raceway in north Austin, but it closed in the early 70's.
I'm not sure the timing for our trip would work. The schedule shows the timeslot to be Friday afternoon but it's 45 min from the Oasis. By the time we got there, it would be 3:00 or 4:00. To make it worth the while, you'd have to take the whole afternoon, which would mean missing the Oasis lunch. Not sure if we'd have many takers if that were the case...