When I wash a quilt, I put it in the bathtub, with the soapy water (quilt soap), agitate by hand, then rinse and repeat until the water runs clear. Put the quilt into a clothes basket. It will be very heavy and you don't want the threads to break.
I have two methods for drying.
1) Lay the quilt on sheets on the lawn, cover with sheets to protect from birds and the sun and let dry flat.
2) Take to studio and pin the quilt square on foam boards to block the quilt square.
I don't use the washer and dryer because of the problems you have had. I also want my quilts (and my grandmother's!) to last as long as possible. I want the wear to be from USE, not from LAUNDERING!
Now - I'll go see what everyone else says!!