Amazon has two distribution warehouses (one near the big airport, the other at Alliance in north Fort Worth) in D/FW. They've been offering same-day here for a few months now.

Just a note: Austin is far smaller than Houston and Dallas

I have a cousin who lives in Austin and he says he's ordered a Mountain Dew from Amazon and had it delivered two hours later by someone in a hatchback. This was about two months ago. YMMV.

Yes, the service is called PrimeNow and it does 2-hour delivery for items commonly found in supermarkets and stores around your area.

This most likely is because the launch locations are physically close to large fulfillment center locations.

Looks like Irving near DFW airport is covered, but not much of rest of DFW.