OK, so an update...

I pulled a couple of the bottom plates. Looks like I didn't drill
through the slab. I assessed this by trying to drill with a non-
masonry bit in a couple of the holes, and they didn't go anywhere.
Most of the holes were about 1.5 inches deep. But the plates I did
pull were certainly moist underneath.

Over the phone, one local place suggested I look at the height of the
first concrete block. He deduced the slab thickness by assuming the
slab was as thick as the buried portion of the first concrete block.
So that;d be about 4 inches. He also didn't want to come out because
he didn't believe I had a problem. He's convinced I need to grade
better and do a better job of keeping water away from the foundation.

The second guy is coming by Wednesday night to physically check it
out. He believes it's condensation wicking through the slab and
expressing itself on the bottom plates. His suggestion was the same as
the first guy's, but contingent on stopping by and actually looking at
it.

Everything is on hold for now. If I have to install a French drain, I
can't afford it. I'll probably pull out the existing framing and scrap
the refinish. Maybe I'll insulate above grade and just make the
unfinished basement as comfortable as possible.

So we'll see... My situation seems to be unique. It's nowhere else on
the Web. I hope this is a reference for people in the future.

Thanks for your help!!!