Followup - thanks for the replies - I removed the bed platform and found (somewhat tight) access to the plumbing. The lines attach to the shower sprayer assembly via two theaded twist-on things and one of these wasn't tightened down all the way. I suspect this was because a section of the line that it was attached to was being pushed on by a part of the wood that makes up the bathroom wall. When the trailer flexes during travel, it probably flexed the threaded join and "helped" work things loose.
I moved the entire section of line to the inside of the wall so it has some room for movement, tightened up the threaded joint and tested - no more leak.
Funny, same thing happened to my previous airstream - same type of joint had worked itself loose. Fortunately there was a (very small!) place to access it. (though in that case, I don't think there was anything putting pressure on the line - just a loose joint)
I'll tackle the sooty furnace next. This is the first time I've run it, so maybe it just needs to be adjusted... Where's that adjustment screw?