I know it's not very feasible, nor common to see winter/spring digs (at least not for me), but I would like to ask if there are some in the US? I know Utah has a few weather protected dig sites for year-round digs, but I doubt those facilities accommodate volunteers. Any ideas?
There are a few. Check the sources listed at Archaeological Digs, linked at this website on the home page. For example, I believe there is a dig in Southeast Asia with Earthwatch related to the origins of Angkor, and there is a dig in Jordan (Tall el-Hammam, or ancient Sodom?) at the Biblical Archaeology Society listing. These are overseas and so they are not cheap, but they take place in the winter.