Like garlic, ginger is best stored cool but not refigerated, or it will rapidly dry out.
It will still dry on exposed faces of course, but you can just slice thinly off until you come back to usable flesh.
It may also start to resume growth. That's no problem, if anything I'd say it's good - it's indicative of freshness and good conditions.
A small pot (you can also find them sold specifically for garlic, and perhaps ginger - sometimes oven-safe too) on the side is ideal, failing that just any ramekin/bowl/shelf/whatever - a pot would just help maintain a stable temperature and avoid light.
I find it keeps for around a month at room temperature, longer cool. You may have better luck in the fridge if it's easier to keep sealed there than cool without.