I harvested my Music hardneck garlic today. Let it dry before storing.
I do pretty much the same. Use my Felco pruners, cut the tops and roots right off.As for curing, I've never washed bulbs. I hang them somewhere shady and breezy, under cover if there's a threat of rain, and when they're dry I trim the roots and maybe peel off the outer layer if I think I can spare one.