Thanks to everyone for their assistance. The job turned out to be even easier than you said (which was a pleasant surprise). The nozzle did, indeed, have a "beard" of unburnt fuel on the outside. I disassembled the nozzle, sprayed everything with carb and choke cleaner, re-installed, bled the fuel line (although I hadnt' disconnected anything) and now we've been operating for 20 minutes without an issue. Since that is a lot longer than it had been able to operate I am hopeful that the problem has been resolved.
Joel