Your Dorothy Hayden azaleas are Robin Hill hybrids that usually stay mounded and compact, little to no pruning required. If you do wish to guide their shape, best done immediately after blooming, and of course it's always a good idea to remove anything dead or damaged immediately so as not to offer a foothold for secondary disease or insects.

Keeping them mulched to conserve moisture and cool the root zone helps, if they are struggling, have they had enough water in the last several months? Not tolerant of drought...

Wait till the weather cools some before attempting to move any 'plantlets' that have self layered. You could move or pot them up this Fall, or even wait till early Spring if your winters are harsh. Use pruners to cut them from the parents, plant no deeper than they were growing while still attached to the main plant.

Thanks for the info. You're right, they have stayed compact all these years with very little pruning and the white flowers are very large, blooming over several weeks in early June. I think I'll wait until next spring to move the plantlets.