A common misnomer with refrigeration is that it somehow preserves freshness. This may depend on how somebody defines 'fresh'. Refrigeration actually contributes to formation of mold, so the suggestion here is to not refrigerate. Yes, some things, mainly things cooked or loaded with preservatives , keep a little longer when refrigerated but generally not so with organic plant material. The main thing that can be done if you are concerned about freshness is to grow your own. As for American herbs, many of them are imported from foreign lands and may have sat too long there before appearing on shelves here; buyer beware.
Not sure what can be done about invisible mold. You might consider a general purpose dehumidifier since mold generally begins with moist bacteria.