Here's a thought...fly to Austin, rent a car (as KSH said, you'll pretty much have to) and drive less than an hour west to Fredericksburg...charming small town, can stay in hotel or B&B, they have a bike shop and do rent. It's a pretty bike friendly place and the guys in the shop are very helpful with routes/maps. The weather should be mild and could be wet, but you never know! I went in April last year and we had slightly cool, cloudy days and one pretty warm, sunny day. Nothing you couldn't ride in. Here are a few links...

general visitor info...
http://www.fredericksburg-texas.com/visitor/index.asp

local bike shop...
http://www.hillcountrybicycle.com/index.htm

new site with popular routes and maps...
http://www.cycletexas.com/

Steph