This morning, I went down to Kleindorfer's and explained my problem. Kleindorfer's is an old-fashioned hardware store. Unlike Menard's or Lowe's, they have more in the way of basic components than just kits; indeed, if you're looking for just one left-handed doohickey, they've probably got it in a bin somewhere. Their sales people are also more knowledgeable (and helpful) than the chain stores.
Suggestion number one: to put screws into really tight wood, treat the screws with beeswax first. Suggestion number two: use a tapered drill bit. Having imparted this wisdom, the clerk took me right to a bunch of things hanging on a wall and took down a No. 8 Screw Setter bit. It would do my screws with no by-guess-and-by-golly, and would drill out a set for the head as well. I asked if they had any beeswax. Of course they did.
So, I have spent three hours this afternoon drilling and screwing, and I'm all but caught up to where I need to be. The screws went in like butter, by the way.
This old dog is learning a really important new trick: On any scratch building project, you know you're going to wind up at Kleindorfer's at last; you might as well go there at first from now on.