I dont know how fast you normally sew ... some people sew super fast ... I never have and the secret to quilting in my opinion, especially a Big Beast ... which is 60" + ... is slow and sure.
Plenty of stops (needle down of course) where you can adjust things, making sure the quilt under the needle and your outstretched hands, which are lying palm down either side of the needle holding the quilt nice and flat and just a lil bit taut.
The walking foot should be drawing the quilt through .. you shouldnt need to either push or pull but if youre going at a million miles an hour, which I dont think is the case looking at the original lines of stitching, they cant possibly hope to keep up.
If you have a speed limiter use it ... and maybe you could try slightly wavy lines instead of straight ... which will be a lot more forgiving ... you can achieve that by going freehand (you could make a mark every 6 or 12 inches or so to aim for) or by using your serpentine stitch which will do it for you
I dont remember which machine you have, whether you have a standard WF setup or have an IDT type thing going ...
if its the latter when you press the button to engage the IDT your machine will grey out a lot of stitches stopping you from using anything but a straight stitch which is why I use my old walking foot a lot of the time so I can use the decorative stitches too.
How embarrassed am I going to be whittering on when I find your using an Old 1930's Singer ???