I had this problem and scoured many forums. Suddenly all the dials became unresponsive while shooting. First noticed playback button didn't work.. then it did ...then it didn't, for good. The only responsive interface was the focus select (center of dial) and the c2, set as zoom to focus to dmf/mf settings.
Menu, Fn, AEL, Play, etc, were not responsive, ONLY in shooting mode. The shooting mode dial was sometimes responsive, and EVERYTHING worked in menu mode (which was accessible when usb connected, or by jamming the menu button in the preview seconds after taking a shot)
A number of other forum threads hinted that the problem was the camera getting stuck in 'half shutter' press. The physical button was not stuck, but the camera was tricked into thinking it was in half-press mode, wherein it was only focusing and none of the other feature settings are available.
Some other replies noted some improvements with wd40 in the shutter button mechanism (WOULD NOT SUGGEST), but hinted that the switch maybe the culprit.
A local repair shop gave an estimate of $565 CAD to replace the "circuit power circuit board", sounded suspect so I took my chances DIY'ing it with a precision phillips head and a spudger.
Camera was off warranty so I gave a go at a proper diagnosis.
ifixit has a guide to remove the back and top panel on the sony a6000, very simple, just keep track of your screws as they are not all the same size. Once the top panel is removed it was clear that there is an excess of lithium grease on the switch mechanism, it's possible some excess has migrated into the actual switch on the pcb causing the issue. Essentially, the pcb switch is contacting in a way it believes the camera is in half-press mode when it's not.
With isopropyl and a cue tip, I carefully dropped some cleaner in an around the pcb switch. Just enough cleaner to permeate the mechanism but not to drip about the pcb. Quickly wiping any excess, I let it to dry for about 5min. I click assembled the panels (without screwing), put the battery in and tested, immediately 80% improvement.
Prior there was 0 response from rear panel buttons, now I could clearly see that it was 'sticking'. I could shoot and the buttons functioned normally most of the time, with some 'sticking' prevalent which was often remedied by lightly tapping the shutter button to jog it fully open again.
I made a second pass with alcohol, cleaning and letting it to dry another 10min before re-assembling the device completely.
I've shot for a few minutes without 0 of the issue. Expect that it might pop up again but I know how to quickly repair it now. I removed some of the excess grease as well around the shutter / on/off mechanism, in case that was potentially the cause of the short in the switch.
One caution in opening the ilce-6000 was a smaller foam o-ring around what appeared to be the camera mic (located under the right side of the hot shoe when front facing) was difficult to replace as the adhesive had dried. I've left it out for now but seldom do video recording so don't expect any noticeable effect.
If you've found unresponsive of or sudden failure of the controls in shooting mode in your sony a6000 , a6300 , or NEX camera, I hope you find this guide and give a shot at repairs. It was very simple and saved me hundreds!