[old_tv_nut]

If you adjust centering, both sweep ringing and video will move together, so this cannot be used to separate the cause. Instead, can you turn the H hold back and forth a bit to change the horizontal phasing? If the picture moves but the rings stay still, they are related to horizontal sweep somehow - however, they still could be multiple things, ringing in the sweep or a horizontal transient getting into the video.

Needing a bigger cap across yoke/flyback sounds to me like something wrong with the yoke or flyback.

Edit: I don't know what the H drive waveform should look like. Can you verify that it's the correct duty cycle? Is there a drive adjustment coil you can try?