I think I finally got it fixed. Completely drained the tank, put a gallon in from my can (fuel in the can was probably 3 weeks old), bled the lines and had the same problem: ran briefly, then sputtered out.
Today, I drained the tank again, put in fresh fuel (purchased couple of days ago.). Bled the lines again - and bled them for much longer than I had done previously trying to make sure all of the old fuel was out.
Fired it up. It was a little slow to catch and run smoothly, but after a few seconds it smoothed out and ran smoothly for 20 minutes before I shut it down.
Tomorrow, I'll give it a real work-out and see how it goes.