Looks to me as if there are a lot of bias edges on those pieces. What would have helped is heavily starching the fabric *before* cutting. (I use a 1:1 solution of Sta-Flo laundry starch and water, which is much stronger than just spray starching.)
If you use a white batting, the white seams will tend not to show through.
Because of the edges you have on this quilt, you would be best off blocking the top. What I would probably do is lay a large flat sheet on carpeting, spread the quilt top on it, and then start measuring and pinning. I would pin the middles first, measure those, and then strive to get the ends pinned down to the same measurements. (You might need to secure the flat sheet to the carpeting with pins or painter's tape so it doesn't move around on you.) Once you are satisfied that it is as straight as you can get it, try misting the top with hot water (if you have a steam iron, you could hold the iron above the fabric and mist it with steam). Put a fan on it to speed drying. Finally, I would mist the top with several layers of spray starch, drying in-between layers, to add stability to the blocking.
I would plan on doing fairly heavy all-over quilting with a fine thread (such as Bottom Line or Aurifil 50wt 2-ply). Lots of the distortion will become hidden in heavy quilting.