I'm actually from Albany. There are 3 Asian markets there: "Asian Supermarket", Kim's Asian Market (never went in this one, but I think it's Korean rather than Chinese), and one more whose name I forget that's smaller but seems to have fresher produce. I never tried buying produce there (we'd mostly hoard Asian snacks) but I wouldn't be surprised if it was more expensive and less fresh than Chinatown simply as a result of the additional shipping/refrigeration required. Anecdotally, no family I knew would shop at those over the standard region supermarkets unless they needed some specific Asian ingredient, whereas in NYC many shop preferentially at the Chinatown markets.