BJ, even on a long-arm, there is some stretching involved. Think about it. If you pull on any piece of fabric it will stretch a different amount in 3 different directions; lenghtwise grain, crosswise grain, and bias. Depending on how the fabric is mounted on the long-arm, you will get a different amount of stretch horizontally and vertically. Add in piecing, and you have a third element of distortion with bias seams!! And the appliqué, with the grain of the fabric often going in yet another direction! So, what's a quilter to do??? First of all, I find that starching helps with the piecing and applique, but I seldom starch my backs unless they are pieced. Yes....this particular back was starched because of all the bias seams, and it still stretched significantly in one area. I'm still learning how to deal with those issues.
Also....you won't wind up with a square quilt if you don't start with a square quilt top! Every step of the construction process should be checked to see that you are making the sides the same length as the center measurements , whether the quilt is a square or a rectangle.
For domestic machine quilting, I pin baste my quilts close enough that any place I put my hand on the pinned quilt sandwich, I'm touching at least one, and preferably two safety pins. On my long-arm, I use Superior's Vanish thread to baste horizontal lines about 4-5 inches apart over the whole quilt.
I can't help you with the hand quilting projects, except to say that they need to be securely basted.
On both my domestic machine and my long-arm, my first stitching is to stabilize the entire quilt by stitching horizontally and vertically in the ditch evenly over the whole quilt. I try to have this stabilizing stitching no more than 8 inches apart, but realize that's not always possible. I also securely SID the borders and the outside edges of the quilt. Then I go back and add more detail stitching over the whole quilt and finally go over it again to do the background fills, trying to keep the density of the quilting even over the entire quilt.
After the quilting is done, I wash my quilt in the washing machine. I first soak it in cold water for about 1/2 hour to remove any chemicals. Remember, I usually use starch, and Crayola WASHABLE markers, and Vanish thread that all need to be removed. Then I spin that water out and run it through a regular light cycle with warm water and Orvus Paste, and usually do one extra rinse cycle to be sure all the chemicals are gone.
When the quilt is finished washing, I block it to get it back into square. I have a vinyl floor in my sewing room that has a grid pattern on it, and can just spread the wet quilt out flat and line it up with the marks on the floor. I pat it into shape and let it dry overnight. My quilts are then pretty square.
Some people will wait until after blocking to add the binding, but I bind mine before I wash it, and so far it's worked well for me. This is not the only way to get a square quilt, but it's what works for me. I hope you find methods that work for you!