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!!!