Fresh herbs, even when dampened, only last in refrigeration 3-4 days. You can freeze fresh herbs, they will last up to 6 months or dry them for up to 2 years of lovely flavor if stored in a cool dark space and sealed from light.
The optimal of fresh herbs is to simply grow them as you would a houseplant and then they are fresh all year long as well as readily and cheaply available.
To dry fresh herbs tie with butchers twine and hang upside down in a closet, in approximately 5 days they will be dry enough to store. Be careful to keep leaves whole for best flavor.