Thanks all for the replies. Furnace repair is over my head, for sure.
I've decided to call a different service company (and hand them a
summary of your responses.) I'll report back here for your interest on
what they say/find/do. Wish me luck. I don't want to be spending big
bucks!

Brian
John Gilmer wrote:
Well, I don't know so much as a tame crow, but it certainly sounds
like there is a hole in the heat exchanger. That separates
circulating air and combusting oil. The fan runs while the heat
exchanger is warm enough to actually heat the air so it (the fan) will
continue to run for a while after combustion stops. The hole might
be small enough or in a place which evades detection. You might find
it yourself by removing the gun, putting a strong bulb in the
combustion chamber and looking in the circulator.

I truly doubt all this.

It's likely that there are a few spots in the combustion chamber or the oil
burner assembly (fuel pump, combustion air blower, ignition transformer and
spark gap) that get a little oil on them that isn't burned. When the
burner is operating there is a strong draft that takes the fumes from the
warm but not burning oil right up the stack. When the burner switches off
the fumes from this un-burned oil will diffuse back into the living space.
This may be made worse if the burner actions puts the living space under a
slight negative pressure relative to the outside air. When the burner
stops, there would be a reverse flow through the burner.

A "cure" might be to bring in outside combustion air to the furnace. A
"box" around the burner assembly that's connected via duct work to the
outside would stop the negative pressure problem and also keep the oil smell
from penetrating the living space.