Certainly, if the planting area is wet you do not want to water. The mulch around the plants is also holding in soil moisture. You should scoop the mulch away and check the soil around the plants to see if the soil is dry before watering them again.
The leaves look like they are infected with a fungal disease, possibly botrytis blight. There is a chance that these begonias will grow out of it. First, pinch off the infected leaves and avoid overhead watering (we know there is nothing you can do about the rain). The white begonia is putting out new growth which is a positive sign. Give them a little more time.