It probably doesn't matter a great deal. Fall has the advantage of cooler temperatures and - usually - more rain plus warmer soil. In Z5/6, however, early spring provides the same conditions except for soil temperature. We probably have another two to three months for root establishment, so I guess I would lean toward doing it now.

Thanks so much! Do you know if azaleas can do well in full sun? The area that I want to transplant to gets afternoon sun from 12-5pm.

I think it depends on the azalea (some are more sun tolerant than others). I'd be careful with intense afternoon sun though.