Is this a raggedy quilt or a normal one with the seams on the inside? I suspect that it is a raggedy quilt where the seams wind up on the outside of the top.
I believe that the batting you are using might be too heavy for the type of quilt. Since you already have the blocks cut and quilt partly sewed, I agree that the best way to go would be to pick out the machine stitching and just tack the center of the block. If it is a large block, you may need to use more than one tack per block.
Next time, use very thin batting or no batting, complete each block, then, sew them together.