Dual Previews is absolutely super helpful, and does indeed help me with positioning! This will help me a lot next time. I know someone suggested it in the previous thread and I should have checked it out immediately. Anyway, thanks!
Using the [left][middle][right][top][middle][bottom] doesn't do any good though, as my screenshot (https://imgur.com/a/ro37xvP) showed. I was however finally able to finish the project by ignoring anything that was close enough and only fixing the positions that were way off. The end result is https://www.youtube.com/watch?v=l9thgJTnbOQ with a good example of what I wanted to accomplish at the 1:37 mark
-0-0-
I also found that the problem seems to be related to re-changing the positions. That is to say: I drag the text to where I want it, then a little later decide I want to move it a little. I now believe this is triggering the problem, the text gets "anchored" at a position and further moving after that causes problems. I also think that sometimes when I drag the text the application lags out a little, thus "anchoring" at an unpredictable spot. This could explain how it seems so random.
-0-0-
I just made a short recording where I show the problem. I open the project file I have been working on, move a text instance, go to another text instance on the timeline that I expected would be moving on its own to show it, saved and closed the project, opened it and voilá: It had indeed moved.
It doesn't move so much that it's easy to spot just with your eyes, because the preview pane is scaled down after all. But when I show the 'So fast' text, hold a piece of paper up to the screen left of the text to see where it is, and when I get back to that text after loading the project again, you will see that it has moved to the left.
https://www.youtube.com/watch?v=xurRVFfFB5Q
The first time 'So fast' is shown is at the 1:27 mark
The second time when it has moved is at the 2:05 mark
Note: the video was just uploaded, it will be easier to see once Youtube has processed the recording to HD