I have had similar "diseases" on various plants and have come to the conclusion that most of them are caused by insects, aphids, sucking on the leaves and their secretions damaging the leaves' ability to feed, breathe and function.. Often the effect is black ...sooty..mould or just sticky leaves and if the leaves are big enough ??ou can wash them. However, Annie, is your azalea large leaved or small?
I would apply a fairy liquid warm water wash.clean as much as possible. Usually azaleas are small rootballed. Is yours reasonably small? I would then move it with as much soil as possible. I would do it now. Hopefully it won't even notice the move. I would not wait for the situation to worsen over the winter
When planted I would apply a sequestered iron feed after watering it in well. Then a mulch......acid bark chips..but not too,much. Too thick a mulch can kill an azalea