Answer: Early fall (early September) would probably be the best time for you, it is maybe a little late in the spring now to do it. If you do it now, do it soon and work just before a rainy spell if possible; trim back the tops somewhat as needed to compensate for roots lost during the digging.
Here are transplanting directions from the Azalea Society of America you might find interesting.
Keep in mind when you dig them that although they are shallow rooted (maybe six inches deep) the roots do spread out wide. You may find that they are intertwined with the neighboring plants as a result. Do the best you can to take as many roots as possible. Good luck with the move!
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