Some of my favorite Broadleaf evergreens are Rhodos, Azaleas, Euonymous, Daphnes, Kalmias, Camellias, Pieris, Leucanthoe, Mahonia, Magnolia. Not all varieties of each of these are evergreen. Nor do all of them do well for me. I have a beautiful tri-colored Leucanthoe that is always struggling. The voles may be eating it and it might do better if moved. I've lost 2 nice Pieris and may try another in the fall in a different location.
I moved a Daphne last year that was struggling, and it virtually tripled in size in it's new location, but now it's suffering. I also moved one Camellia last fall and it overwintered and bloomed really well, but now it looks like it's become a victim of this drought. There's one Kalmia that was in too dry and shady a spot, so I moved it two springs ago. At first it really responded well, but now it's suffering. This drought has been very hard on the shrubs.
I also have some evergreen Viburnum Prague. Another evergreen favorite of mine is Skimmia. One of mine now is under attack by a stubborn case of scale.