I had a friend, he's been gone for several years now, that used to go out in the woods and take cutting of new growth and pot them for resale. Not sure of his exact methods, we talked about it once. I believe what he did was cut the new growth from the tips and scrape the cutting with his knife and then dip in a product called Root Grow. He then placed the cutting in potting soil and kept them watered until they took root. I do believe he said that a lot of cutting didn't make it, but many did. He would grow them out for a year or two and then sell them to individuals looking for landscaping plants. Never tired it myself. I prefer the naturized bright orange plants to the more popular, and becoming more common, hybrid varieties.
I have never attempted to wrap a limb in peatmoss to encourage root growth. My Uncle does something similar with apple trees. He will take a sandwich bag and slip over a small stem, then fill with potting soil, add water and seal the bag. The moisture in the bag provides something for the stem to root in and the tree keeps the stem fed. He has had very good success using this method. I suspect this method would work pretty well with Azaleas. I have taken grape vines and placed a section of the vine in a bucket/pot of soil. The vine will root in the soil and then you just cut the vine away from the main vine. Roots are already in the bucket growing and don't have to be disturbed. Pretty much works every time. Makes transplanting much easier to.
I have never attempted to wrap a limb in peatmoss to encourage root growth. My Uncle does something similar with apple trees. He will take a sandwich bag and slip over a small stem, then fill with potting soil, add water and seal the bag. The moisture in the bag provides something for the stem to root in and the tree keeps the stem fed. He has had very good success using this method. I suspect this method would work pretty well with Azaleas. I have taken grape vines and placed a section of the vine in a bucket/pot of soil. The vine will root in the soil and then you just cut the vine away from the main vine. Roots are already in the bucket growing and don't have to be disturbed. Pretty much works every time. Makes transplanting much easier to.