to me, that looks like it's maybe a pulldown or interlacing issue. maybe you accidentally rendered out the moving text with pulldown or interlacing, and then are accidentally adding it again when you render out of premiere?
if you go through the frames 1 at a time, slowly, does it not move every 2 or 3 frames?
you might want to keep all pulldown or interlacing turned off unless you are outputing to something that requires it, and then only put it on that particular render.
if that's even what the problem is. if it's not that,