i wash old quilts in the bathtub because that seems to me to exert the least amount of 'pull' on them. i use the sheet trick and the towel trick, but when i take it outside, before i spread it out, i carefully step on the rolled up towel to squish out all leftover water. that makes the quilt much lighter to handle. then i lay it out on a clean dry sheet, in the shade, squaring up as well as i can. with less water, it's much easier to square. when i put the second sheet on top, i pat as i go to flatten it out. i try not to pull at all. i think that would tear the old fibres. then i hope that it won't rain. when it's almost dry, i bring it inside and find a safe place to let it finish drying. why push my luck? by that time it hardly weighs anything, so if i have to, i can put it to dry on the guest bed. if not, then it goes on the carpet.
btw, i never move it without rolling it up in the bottom sheet first. then i carry the bundle like a baby. that also relieves the weight..