Yes, cottony hydrangea scale is a pest seen on hydrangeas.

Did not save the link so I am trying again...

Here is a link that might be useful: Hydrangea scale post

Euonymus scale is pretty much specific to euonymus. Removing the plant should remove the problem. Like pretty much any other woody plant, hydrangeas can get a scale - in their case, cottony hydrangea scale - but it is not the same as the euonymus scale and the two are not interchangeable.

Yes, you should identify which type of scale insect you have. Euonymus scale does not drift towards too many other shrubs but cottony scale can infect both shrubs.

I love the suggestion of getting rid of the euonymus, gardengal48! When I bought my current house, I had a few Euonymus that looked nice in green and yellow leaf colors but where a powdery mildew magnet. I decided to pull them out as much as I wanted to keep them and still remember the thick cloud of PM that arose when I finally got them out of the ground. Lord, oh Lord!!! I still want to buy those shrubs when I see them at nurseries but they get too many problems....

Here is a link that might be useful: Cottony scale Info

Thanks luis pr and gardengal48. Your help is very much appreciated!