Dear Wendy and Tolouse:
I'll have to do some more experimentation to be sure, but it looks like the Illustration thing was exactly the problem!
I did not have Illustration selected, but the combo box showed "Custom", and it is possible that Illustration was the last thing selected before I 'customized' the settings by changing the brush. When I followed Wendy's advice and change the combo box to show "Oil Painting," the problem seemed to disappear.
I have only tested with one image, so I'm not 100% sure, but I am very hopeful that the mad slasher has been put to rest for good.
Thank you both!