Ended up removing my carburetor, cleaned it, and it now runs perfectly.
An experienced small engine mechanic told me about eighty-percent of the Honda generators they get in for repair only need their carburetors cleaned and that is really all they ever require, other than routine maintenance.
So, I took a chance, took my carburetor off, cleaned it, reassembled it, and it now runs like new when I purchased it twelve years ago.
Also, while at it, cleaned the spark arrestor, which did not appear dirty, put on a new air filter and new spark plug.
FYI, I learned the carburetor has a handy, easily accessable drain screw to drain the carburetor gas before storage. I did not know it was there. I
Again, thanks everyone for their recommendations and help.