My fabric almost always goes in the dryer, unless it's small pieces...those I iron dry. I dry my fabric completely in the dryer, then iron it and I"ve never had a problem.
I found a site once with great directions for folding and storing fabric, but I guess I forgot to bookmark it. I'll try to explain it here.
Fold the fabric selvedge to selvedge and press. Then fold again by bringing selvedges to the fold. I don't press again at this point. Measure 9" from the end and fold, continue folding by flipping the folded section until you get to the end. It's easy to tell how much fabric you have by counting the folds...four folds is one yard.
It takes little extra time to do this but it really makes storing fabric neat and easy since each piece about the same height, just a different thickness depending on the yardage.