I have left it running for 5 minutes or more while I am trying to get the pump to prime, but no longer than that. I never smelled any burning, although the pump did get pretty warm.
Based on the age of the boat, and the fact that I've never changed the impeller, I'm guessing the impeller has gone bad. I ordered a new green impeller and will crack the pump to check it out and report back.
Thanks for all the help.