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I would be concerned about the gingers rotting if the ground is always moist.
The gingers just naturally work themselves up out of the soil like an Iris.
You could try heavy mulch to keep them dry.
Leave a few and dig up a few just in case.
Next year amend the soil so it drains faster and you can leave them in the ground.

We are ginger nuts here in New Orleans and its ... well we are below sealevel, our soil tends towards thick black clay and we get lots of rain. With that said we don't get the cold like you do and that along with the wet may cause rot.
I also have them growing at my place in Ms. z8 which has a sandy soil and they winter there with zero mulching. 'Course I don't have the wet soil you do there. The butterflies are very hardy from all I've ever heard and have always been reliable for me.
Mike