Well, the tank has been drained. I threw some IPA(don't worry, good, dry stuff) and a gallon of gas. So far, it's running a lot better but I'm going to run it empty before I refill the tank and hopefully things will be back to normal(although I probably should drain the carb float bowls and such).
BTW, I've not run it close to empty in 2 years of ownership(I didn't know where empty truly was on the gas gauge before today and didn't want to chance running out), although I'm guilty sometimes of letting it sit for a while with 1/2 to 3/4 of a tank.
I don't know if it has any relevance, but this is an "open" system without any vapor recovery. The gas filler is on the back and runs straight down into the tank, and the cap really is just a cover as it has to allow air into the tank and vapor out. After I finished with the head overhaul, I had it out for an hour or so in a rain storm-something I don't generally do, but I was anxious to get back out on the road and do some road tuning. I wonder if I picked up some water there. If so, I might need to look into a new gas cap also.