First, I replaced the washer in the shower head. No luck.
Then I bought a new shower head and installed it. No luck. It also leaked a drip at a time out of the shower head itself.
Interesting: Even with no shower head installed -- the drips occur directly out of the arm. One small drip at a time --- about every 4-5 seconds.
I have no idea what the arm is connected to and/or how the on/off handle may be involved here.
I have no skills whatsoever --- so if I call a plumber --- what is the expected level of effort (i.e. cost) of a fix here?