Just to update this....
I could find nothing wrong, so I perservered attempting to burn discs. Of the pack of 15 Sony DVD-R, the first 12 failed. The last 3 worked fine.
I decided to go ahead and open the the second pack of 15, rather then take them back, thinking maybe I just had the bad luck to run into a bad batch. Of the first 3, 1 failed.
I'm of the opinion something is wrong with quality control at Sony. I've picked up a 50 pack of Memorex. After I get done with the Sony, it will be interesting to see how the Memorex perform.
Maybe, as someone here suggested, the burner is just wearing out.
Thanks again for the suggestions, peeps.