Thanks for that advice. I've been choosing the contour of the top opening, not the bottom face as you suggested. I guess I should have read the manual on Engrave.... oh wait, this forum is the manual...
Anyway, my entire progress slowed to a near halt because when re-selecting the dots and dashes for Engrave, each dot or dash would take about 10 seconds to register (highlight). This made work impossible. Then, I made a copy and set out to do a complete re-layout. In doing so, I remembered that there is a specific bug with bodies that have a large number of extrusions in them. In general, when there are a large number of extrusions, like holes, etc., in a body, any new operation causes the program to recalculate everything globally and performance goes right into the toilet. So, I removed the extruded bodies (holes) from the body and setup the engrave from the sketch contours. Amazingly, I got a fairly decent tool path:

Now this is very close to what I would expect for most engraving operations. Its also the second time that I've had a major performance hit with multiple extruded features.
I can only wonder what the experts think about a job such as this. Should you extrude/inset engrave features into a body, or is it recommended NOT to extrude them and proceed directly to Engrave from the Sketch contours? I think I've answered my own question but it sure seems like some guidance in the form of a manual page would have saved a lot of time.
As an aside, I would say that the drawback of not engraving over an extruded model is that you lose some of the 3d preview in simulation mode. Not such a bad tradeoff.
That said, I'm still a big F360 fan, and remain truly impressed with the product.