Potatoes keep about forever if they're fresh; (well, at least a few months) if they're kept cool. Finding them at a farmer's market or local farm stand is the best deal. The fact that they about rot after a couple weeks at home is an indication of how old they really are.
In NYS, there's two main areas that grow potatoes; highland and muck. The highland potatoes, which are grown in higher, drier conditions store much better. Some varieties store much better than others as well. It's worth looking into.
I believe that both of you have hit it on the head, keeping them cool. I live in TX, I thought I had them stored in a cool enough place but maybe not. Plus the fact that the potatoes sold here are probably elderly storage potatoes.
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