quiltlady, you need to realize that quilting binds the fabric to the batting, at which point the batting limits how much the fabric shrinks. The fabric in that one block you washed had not been stabilized by quilting to a batting. If you were to take another block, quilt it to batting and backing, and wash it, the brown fabric would not shrink like that. The batting would prevent it from shrinking so much.
You would not want to tie this quilt. Tying would not be enough to limit the brown fabric's shrinkage. When quilting, you also do not want to quilt lines 10" apart. That would give the shrinkable fabric too much space. As long as your quilting lines are a reasonable distance apart (say, 4" or closer) and you quilt through the sashings several times, your quilt will turn out fine.
The one thing I would suggest is probably not using a 100% polyester batting, as polyester does not shrink at all. Other battings shrink a little in the wash, which helps camouflage any uneven shrinkage of the fabric.
An all-over meander of quilting lines, with lines no farther apart than 4", would look nice and would ensure that all of the brown fabric is tied sufficiently to the batting.