Well, folks, I'm about to give up on the edger, for the time being. I
have taken it completely apart, cleaned with carburetor cleaner and
compressed air (gently), and still it won't run (unless I squirt gas
down the spark plug hole, and then for only 5-10 seconds).
I would like to perhaps buy a carburetor kit, but Sears wants to sell
the entire carburetor for ~$70.00.

I was wondering if anyone knew of a Craftsman to Briggs and Stratton (I
think that is who made this engine, but I'm not sure) cross reference
resource? I have the Craftsman engine serial number, and if I could
contact a B&S retailer, perhaps they have a carburetor kit that might
work for this carb.
Thanks again for everyone's advice and suggestions, Richard