Answer: This could be a fungal infection. I would suggest removing the affected leaves and put them in the trash. Do not handle the plants when they are wet or damp with dew. If more occur, consult with your county extension and/or the retailer to get a more specific diagnosis and then knowing that, decide how to proceed.
In the meantime, watering is done, if needed, to supplement rain. The soil for both of these plants should be organic and humusy and evenly moist yet well drained, meaning not sopping wet or saturated. To know if you need to water, dig into the soil with your finger. If it is still damp you do not need to water yet. When you water, apply the water slowly to the root zone and avoid wetting the foliage. Your goal in watering is to keep the soil damp like a wrung out sponge. It is better to water slowly and deeply less often than to sprinkle lightly every day. After you water, wait a few hours and dig down to see how far the water soaked in; sometimes this can be surprising. For an azalea -- these are shallow rooted plants -- you need to keep the top six inches of soil damp, for a viburnum it should be deeper. You want to encourage deep roots because that is where the soil naturally stays moister longer.
Using a layer of organic mulch two to three inches deep over the root area will help keep the soil moist, hold down weeds, and also help feed the soil on an ongoing basis as it rots down. Good luck with your plants!
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