The issue described below is solved, but I post it anyway since it might help others.

Taking a picture resulted in losing video signal for a short period and moving the gimbal to the most right position. Then, after a while, the gimbal slowly recentered.
There was no impact on stability of the plane. It only affected the gimbal.

I have recalibrated censors and changed several photography related settings, including disabling the mechanical shutter. No result.

Then I saw a post from @Droneczar in a topic about the HDMI output not working. He also had issues like I had and suggested to go into the DJI GO App camera settings and hit Reset All Camera Settings.

It looks like the issue is gone now.

Thanks for sharing!