If the drooping leaves are all on the same branch:

1) Terminal buds and leaves turning brown and leaves wilting is symptomatic of Phytophthora, Botryosphaeria, or Phomopsis dieback. All three diseases spread rapidly and may threaten an entire plant. They are more active in warm summer conditions.

i) Phytophthora causes the central vein of a leaf to turn brown and the discoloration extends to the petiole on tender new growth. The infections spreads outward from the midrib tissue and the leaf wilts. Infections are more severe on azaleas. Some varieties of rhododendrons are vulnerable (Chionoides, Catawbiense Album, Nova Zembla) and some are resistant (Roseum Elegans, Scintillation, PJM). Control of the disease is difficult. Since the infection goes from the roots to the tips, when you see the symptoms it is too late. To prevent it, use a raised bed with lots of sphagnum peat moss. Prevention with fungicides and careful control of exposure to high humidity may be practical.

ii) Botryosphaeria causes leaves to turn dull green and then brown and roll and droop. Cankers form on branches which may girdle the branch. Sanitation and applying a fungicide such as metalaxyl (Subdue) after pruning my provide some control.

iii) Phomopsis symptoms vary from leaf spots to chlorosis and then browning of leaves which then wilt. Browning streaks extend down the stem to a wound. Fungicides such as metalaxyl (Subdue) should control an outbreak. Sanitation and applying a fungicide after pruning may provide control.

2) Drought can cause entire branches or entire plants to die. We have had several years of drought here and we observe that if rhododendrons and azaleas are not watered during a drought some plants will die, but others will just have one section of the plant die. It seems to be the plants way to conserve what little moisture it has.

3) Bark split is most commonly caused by an early autumn frost while the sap is still high in the plant, or a late spring frost when the sap has already started to rise. For this reason it is dangerous to feed nitrogen to a rhod
odendron or azalea that could stimulate growth through to autumn. Don't use nitrogen after mid-June. When the bark is frozen, sap cannot get through and the bark splits. For this reason always keep mulch away from the trunk of plants. Bark split damage can be treated with grafting wax to prevent fungal and insect damage.

4) Ends of branches die when rhododendron or azalea borer larvae tunnel in a stem. This affects the portions of the plant away from the roots from where the borer larvae is in the stem. Borers have done their damage when you see the wilting and usually it is best to cut off the affected region. The requires removing the damaged branch from the hole where the eggs were laid at the base of the dying branch to remove any larvae before they become adults and infect more plants. The parts that are cut off should be destroyed to kill the larvae that are in them. Borers are prevented by following a spray schedule (timing is very critical) for borers with chlorpyrifos or lindane. Your county agent can help you with this information

From your descriptions above the one that makes the most sense is bark split - everything else seems to be ruled out as the leaves have been drooping for weeks but there is no sign of browning, the weather here cannot be called warm, I cannot find any sign of borer holes, and it rains at least twice a week torrentially.

Regarding the bark split - it seems to have occurred to all of the branches - not only the ones that have drooping leaves. Does this make sense? Once treated will they revive? Thanks for the thorough information so far!

Maybe, Maybe not. It is difficult to keep bark split from causing the eventual demise of a plant. The open bark not only kills the area under the bark, but opens the plant to infection. I have a dwarf plant about 30 years old that died of bark split this year. I am not sure when the bark split occurred. I didn't catch it until this year, but it looked like it had happened several years previous. It was coming along fine until this spring, when it gave up.