Azalea lace bugs. They are very destructive. Check the underside of the leaves.

Fighting them is not easy because you have to spray the leaves from below.

https://agrilife.org/etg/2012/08/12/azalea-lace-bugs/

id probably go with a systemic remedy ..... theses are sucking bugs.. so if you apply something the plant sucks in.. it then kills the sucking bugs ...QED ...

http://extension.msstate.edu/content/azalea-lace-bugs

and if you dont prefer that.. read all these and come up with your best alternative:

https://duckduckgo.com/?q=Azalea+Lace+Bugs+systemic+control&t=ffcm&ia=web

if you have never sprayed anything... we could offer some guidance on that.. one thing for sure.. never in hot sun ...

i also note.. the one link i reviewed ... indicates they attack stressed plants ... it might not hurt to figure out why you plants are stressed.. and stress has nothing to do with fert.. so dont fert for that reason.. besides any decade old plant never needs fert .. most likely a water issue ...

and to be sure.. these leaves will never heal.. look good again ... it will be a matter of solving the issue... and then having lush new growth cover the offending leaves..until they naturally fall off ...

your county extension office might also offer guidance.. in case this is some local plague ... bug populations ebb and flow over the decades... you may never see this problem again.. if you solve it ... or it might be back next year ...

ken

Thank you. That is all helpful information. Is this something that I can treat, but the pests will probably come back year after year? Or is it a pretty high recovery rate once treated and eradicated?

We are also considering pruning some of them almost down to the ground, because as you can imagine after 40 years, some of them are a bit out of control. Do you think this would help as well?

You should get some local info on that (e.g. your local extension or independent nursery).

Generally it's not recommended to hard-prune azaleas lower than 12", because they may not come back, especially if they are old and already weakened by pests/diseases.

But, as usual, it all depends on your climate, plant variety, available nutrition and general vigor of your azaleas.