Other things I would do:
First take out the last seam on both rectangles to return it to 4 quarters.
Then iron all the seams apart - I would first fold the paper along those seams and remove JUST the outside paper so I could get a flatter ironing.
Then I would tear away the very centre paper of the segments which have the outside sewing line printed on, and also a larger area of the inner segments which do not have an outside seam to be sewn. This is all to reduce bulk. If necessary I would redraw the sewing line with an erasable marker. Any dog-ears can come off too lol.
Then on to the re-sewing. As Tartan mentioned, don't sew across the very centre. However DO start your sewing at the middle and sew towards the outside - that way the centre will not have any chance to shift. A couple of reverse stitches will anchor the seam end.
If your outer join matches are a problem, then you could perhaps baste the joins before doing the whole seam, to keep them aligned.
If all else fails and the fabric is too stretched to match up easily again, then I would fold back all the joining seam papers (leaving the other paper in place), match the joins up with a bit of easing from the unstreched side, and then steam and block the unit square afterwards, using the remaining papers as a reference.
Good luck, hope it comes together for you :-)
cheers, K