Apparently, the malingering smell of exhaust in the passenger compartment wasn't just my imagination.
While replacing the motor mounts, I'd discovered that the exhaust downpipe had come loose and was hanging on by only one bolt. Furthermore, the gasket was severely eroded.
I decided to replace both studs and nuts. I was able to back out the old studs (actually, there was one stud, and one mismatched bolt) after a good hour-long soak in WD-40. (I picked up some PB Blaster later in the afternoon, but didn't need it.) I installed the new studs after dressing them liberally in anti-seize.
After unsucessfully trying to locate an OEM gasket for three days, I stopped by my local muffler shop and asked them if they could find something. The fellow there found one on his Wall-of-Gaskets and even gave it to me for free. Free! (In this day and age!!)
Back in business!