I make drawstring bags out of used curtains (for ladies in abuse shelters). The fabric is usually heavier than quilting cotton, lasts a long time, and you can get some very "expensive looking" fabrics. I even use the drapes that say dry clean only. I just wash them in cold water after deconstructing them and before cutting and sewing. I believe often times the dry clean only warning is there simply because the drapery fabric and lining fabrics would shrink at different rates if the completed drapes were washed, and you'd have ruined drapes because the lining would be longer than the drapes, or have puckers, etc.
You don't have to use recycled drapes for your gift bag, of course. Take a coupon and check out the drapery fabrics at JoAnn's!