I've read that storing Ginger in wine vinegar or vodka preserves it. Is there something non-alcoholic that will do the same thing?
Vinegar is non-alcoholic, and my suggestion would be a brine of some sort, essentially pickling it. Alternatively you could just dry it out.
I should also add - Ginger is a root, and can generally be kept fresh in a cool dark place (ie. a root cellar) for some time.
I appreciate these answers and comments, but I found some help at a different website. As it seems to answer the part of my question about cooked ginger, I'm posting it as an answer, but I've never done this before, so please correct me if I've posted it incorrectly or if it should have been a comment.
The Frugal Chef http://thefrugalchef.com/2012/11/candied-ginger/ has a recipe for cooking ginger in a simple syrup of equal parts water and sugar. Once it's been cooked, she says, "You can leave it in a closed container in the refrigerator for up to a year."
I'm disabled and can't cook very often, so this would be perfect for me. Have any of you tried this? Thank you.
Sugar syrup would probably work as well if the sugar solution is fully saturated plus some extra sugar to compensate for any water pulled out of the ginger.