Thanks to all that responded with good tips and advice. In response to a few thoughts mentioned here, the gas being used (non ethanol) is fresh and properly mixed and the spark plug and air filter have been recently serviced.
The more I have read has lead me to think the problem is the chain oiler. I looked at a couple of youtube videos regarding replacing the pump and associated worm gear. It looks like a fairly straight forward fix that I will attempt to do myself. The parts aren't too expensive, and if I somehow screw the repair up, I can always take everything to a qualified technician for help. Don't ask me why I know about this option. Hint, past experience.
Anyway, thanks again for the responses. I'll give a final report on how this all works out. If all else fails, I can buy a new saw.