As somebody who was accepted as a transfer student to UT-Austin having withdrawn a class, I can tell you that they do accept people with Ws.
However, having said the above, I would say that your circumstances are unique and require additional explanation. UT-Austin gives you plenty of opportunities to write during the application process; take advantage. Your GPA has gone up, so the trend is positive, and that's what they want to see.
I wouldn't worry too much. Keep the grades up. It's never to early to start thinking of how you would explain your early mishaps.