Acording to the User Manual the WX50 can not record date while in Movie Mode.
If it is doing that, the camera has a defect.
If you purchased the camera within the past year, it should be under warranty.
I suggest you contact Sony Service.
Look toward the end of the page under "NOTES"
I have a Sony DSC WX50 digital camera and I can't get the date stamp to turn off on the video.
I have turned it off for still photos, but can't find the option to turn it off for video. It is not addressed in the manual. I am using AVC HD setting, if that matters.
Any help would be great. This is driving me crazy.