As I recall from the ones I grew in Asheville, they do sorta die off at the tops and then I dig them. Rinse well, store in a shady place for a couple of weeks to dry out some. I put in a few here last fall and they are growing quite well now. Same for my fall planted garlic.
My garlic didn't do so hot, but I planted really late too. But there are fields close and when harvest is finished I will go and glean. They leave so much behind. I raise sweet onions in my raised beds too and they are looking great. Can't wait for those, mmmmmm!!