I believe I understand the cause of the problem - your Sequence is Progressive, while the Export is Interlaced (Upper Field First).
Are your source video clips 1080i or 1080p? I see you are using 1440x1080, which suggests maybe the source is HDV tape? That is typically 1080i.
I might also question whether the 29.97 frame rate is correct - which country are you working in? Most countries other than US and Japan use PAL video, while 29.97 suggests an NSTC country.
Please right-click a source clip in the Project Bin and select Properties - what are the specs of the video?
When exporting to H.264, you almost always want to use Progressive rather than Interlaced, because computer displays are not interlaced and do not display 1080i video very well. You want 1080p or 720p files for computer/online viewing.
Just watched your video again, which I had not seen in several days, and noticed that the background is all still images, no video. If you are not using any video, then why the 1440x1080 sequence, with 1920x1080 export? Would be better to use 1920x1080 consistently throughout the process then.
Just seems that there may be multiple workflow issues happening at once, adding to the poor output quality.
I've worked my head around the problem and tried saving using different settings (given that I cannot change the sequence settings, right? and so I'd have to start from scratch).
I finally discovered the Match the sequence settings checkbox, but even allowing for the not optimal 29.5 fps in the sequence, I still have a jerky motion all along. I attach the screenshot of the export dialogue.