How can I measure water/moisture content within a wall cavity?
I've used a moisture meter to do roughly this. My meter is the pinless kind. Some years ago, we had a contractor who did a less than perfect job on a bathroom remodel. (Ok, they left behind leaks in something like six separate places.) With the meter, I was able to tell where water was a problem, and also know when we had finally gotten all the leaks repaired.

I would not expect the actual meter reading to be highly accurate, since these meters are usually calibrated for wood. But the meter will indeed be able to tell the presence of moisture, and to differentiate between dry and wet. The problem is, these meters can only read to a certain depth, typically a half inch or so in wood. I'll look tomorrow to see, I might be able to test mine under similar conditions to see how deeply it can read.

Edit: I tested my moisture meter. I have a 12+ year old Wagner L609, an inexpensive pin-less meter that works nicely for wood samples. It is not tremendously accurate, but it does show moisture differences well between samples. Newer models might read more deeply than did mine.

The first test that I ran was to test a piece of 1/2 inch thick drywall, then move the flat of my hand up to it, until I saw a difference. By itself, the drywall reads at roughly 5-6%, my hand by itself reads off the top of the scale. When my hand approaches to within roughly 1/2inch, the meter sees a difference, jumping up by about 3-4%. It could also recognize the presence of my hand on the other side of a 3/4 inch thick telephone book.

Of course, this is a poor test for what you are doing. Suppose that moisture collects on the inside of a vapor barrier inside the wall, separated by over an inch from the interior wall surface? Were this to happen, the air space between the two surfaces will be maintained at a high humidity. In turn, this will significantly raise the moisture content of the interior wall surface.

I next did a test where I showed that drywall faced with a somewhat moister environment on one side would then easily register several percent high in moisture content.

This last test convinces me that you can indeed check for elevated moisture content of a wall cavity using a non-invasive and inexpensive moisture meter.