Answer: Azaleas would do much better in morning-only sun or bright dappled light all day rather than hot afternoon sun. The Leyland cypress may also be sending roots into the adjoining area and causing the azaleas problems that way as well.
For a hot afternoon sun location with potentially drier soil you could consider daylilies, purple coneflowers, sedums, or sturdy shrubs such as golden barberry, spirea, or forsythia.
Whichever you plant, keep in mind that the soil will need to be kept evenly moist (damp like a wrung out sponge) while they become established and rooted. If they are planted too close to the Leylands it may be quite difficult to succeed with almost anything due to the root competition. Also, it would be better to wait until early September to plant rather than try planting now in the heat of summer in a sunny spot.
In the meantime though you could transplant the azaleas to a shadier spot now as they may not survive if left in place.
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