You could probably have avoided most or all of this problem by heavily starching your fabric before cutting and piecing. Starch stabilizes fabric so it is much less likely for bias edges to distort with handling.
What I would do at this point is "block" the entire quilt using spray starch. Sharon Schamber has videos on Youtube that show how to block squares that are mis-shapen. You can do the same thing on a larger scale with the entire quilt top (omitting the ironing). Basically you are going to need to pin and spray until you get the whole thing squared. You probably need to start with a king-sized sheet pinned to a carpet. You would have to lay out the finished dimensions of what you want this top to be (I'd probably use painter's tape) on the sheet. Lay out the top, pin the dimensions that fit, then spray to relax the fibers. You can stretch areas that need to be stretched, pin, then blow dry. For areas that need to become smaller you can do the same thing, seeing if you can shrink the fabric a little.
You may not be able to make the top perfectly square with this method, but you should at least be able to reduce the difference between the two sides by an inch or more so the out-of-squareness will be less noticeable. Depending on the pattern, you may also be able to take in or less out some seams along the edges to minimize the difference.