If you're willing to invest the time, money and effort, raised beds on top of the unsuitable clay soil are an option. Dense shade and strong winds are much less easily dealt with as is root competition from the existing trees, The latter can be a real issue depending at least in part on the tree species involved. Thinning out some of the trees and/or limbing them up might help greatly.

As for varieties, any of the species rhododendrons native to southern areas such as carolinianum, catawbiense, and cumberlandense and their hybrid offspring should be suitable. There are also some new hybrids bred specifically for the south which involve the Taiwanese species hyperythrum. Meadowbrook Nursery has a good selection. There many, many evergreen azalea varieties which should be suitable , but many of these may suffer extensive leaf damage from winter sun and wind.

All rhododendrons and azaleas have very shallow roots. They also have a tendency to develop root rot if the roots are kept too wet in hot weather. For these two reasons, it is best to keep the azaleas in an area with partial shade, excellent drainage, and a good mulch layer. The mulch does two equally important jobs, it conserves moisture in the soil and protects the roots from hot weather.

Here is a link that might be useful: How to grow rhododendrons and azaleas