Does anybody have a great way of organizing and hanging a scarf collection that doesn\’t involve folding?

Absolutely! I take my scarves – and I have plenty! – and I drape over a hangar, much like you would a pair of pants. I may have to fold once, but not all the time! Another thing I found was a hangar that had 3 inch hoops attached in a type of macrame (check out Ikea). The scarves go through the hoops, but you will still have to fold them or pull them through loosely. This may wrinkle them still, but not always.
You could invest in a quilt stand and use that to drape your scarves over – it is large enough to handle even your widest scarves! I would suggest you organize them by how you wear them – least used on the bottom, most used on the top.
I hope these are useful to you!