Answer: It does sound like you have lace bug damage. They feed beneath the leaves, sucking the juice of the plant which causes the white spots on the top surface. The black spots are their fecal matter. There are a number of products which will control lace bugs.
Insecticidal soaps are among the least toxic option, although good coverage of the bottom surface of the leaves is important for effective control. One or two repeat spray applications about a week or two apart will likely be necessary. Other, synthetic pesticide options are also available. Your local nursery professional can assist you with choosing another option if you wish.
Your description of the overall plant appearance suggests that perhaps some other problems are also involved. Make sure the plant is getting adequate watering, but yet with good drainage as azaleas deplore wet feet.
Thanks for the question and good luck with your azaleas.
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