It sounds like you need to turn of video stabilization, which is what adds a pronounced jello effect.
The K-1 does not have continuous video AF. The only AF available while recording is a slow contrast-based system which is activated whenever you press the AF button on the back of the camera.
While not recording, it's possible for face detection to be on. To disable this, just change the AF setting via the control panel to something else.
More details here:
Pentax K-1 Review - Video Mode | PentaxForums.com Reviews Adam
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