What I would do is, Remove them form their husks as soon as they start to split open, lay them out on a shallow tray to dry in an airy place. When you are saticfied they are dry transfere them into baskets or net bags,again in a dry airy place.
The fresher you eat these the better as some of the sugar content will turn to starch when stored for a long time, so yes small batches kept in the fridge may prolong the sweetness for a while. Good Luck Tbird+
I have generally kept mine in a wooden bowl in the living room but this isn't terribly satisfactory because the tend to dry out quite quickly. I haven't seen any of this season's on sale yet, but this autumn I'm going to try experimenting keeping them in an airtight bag in the fridge. I suspect anywhere which is warm and dry is not going to prolong their edibility so perhaps a shed or garage would be the best place.
They don't last that long in our house because we pick them from our own tree and they are really sweet and moreish This is the first year we have had a good crop it could be because the grey squirrel ppopulation as been culled