Your strawberries have gray mold (Botrytis cinerea), also called botrytis blight. As you noted, it is a fungus that acts quickly to rot ripe fruits. If you look closely at your plants, you'll probably see some infected blossoms too. The disease is most common during cool, humid weather. Crowded plantings and rain and splashing water enhance its spread. Blossoms and berries become infected directly when a healthy bloom or berry touches an infected one, the ground, or a dead leaf. Pick and destroy infected fruit to get the source of new infections out of the patch. Don't leave the rotten fruit in the patch! Mulch plantings with clean straw, pine needles, or other mulch to keep fruits off the ground. Keep proper spacing between plants to promote air movement and drying. If necessary, spray with a protective fungicide that's safe for edibles.