I'm having the exact same problem with my Entune Premium. My 2012 Corolla had no problems at all playing random songs from my MP3 player but the 2014 "forgets" what random means after the car has been turned off then on. After entering Browse mode, it takes a very long time before the song list is displayed where I can choose the Random option and fix the problem.
But on the brighter side, Pandora plays more reliably. It takes a long time for the stations and track information to be displayed, and I wish the display would default to the track information instead of the station list. But once it's going, it doesn't cut out nearly as much and oddly enough, it seems like its choosing a more varied selection of songs. On the 2012, I'd get the same songs over and over.
This forum doesn't seem to have much activity so I don't know if an answer will be found here, but I just thought I'd add my comments in case Toyota is paying attention.
Thanks -
Lisa