TurboTax did not go "totally online." They still offer the desktop software, which is what saves a .tax20yy file on your computer. The only thing that changed is that you get the desktop software as a download instead of on a CD. (And you can still get a CD in addition to the download if you want it, for an extra $10.) Once you install the desktop software from the download it's exactly the same as the software that you used to install from a CD.
The desktop TurboTax software and TurboTax Online are completely different products. If you use TurboTax Online you do not have anything installed on your computer, and it does not automatically save a .tax20yy file on your computer. (You can download the .tax20yy file to your computer if you want it.)