Answer: Plumbago is usually trouble free so there is no "usual suspect" you are right. Based on your description I think you might be seeing evidence of a cottony scale insect problem which eventually does cause yellowing, the stickiness could be caused by the sucking and the black spots could be mildew -- and there would be no webs as with spider mite. Without seeing the plant however it is hard to say for certain. Here is a photo, note the white dots ...
Scale can be treated in the crawler stage with insecticidal soap or can be sprayed with horticultural oil to smother it. I'm sorry you are having trouble with your plant.
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