This is EXACTLY what happened to my K2 after it sat for 5+ years following cancer I had in 2001.
I chased it for almost a month: I was getting 3 fouled sparkplugs in less than 5 miles of riding, over and over, with the last one fouling from the others missing, by 6 miles. It took me 3 sets of plugs to get it to work and back, 11 miles total distance.
After lots of things, I discovered that the emulsifier air passages were fully blocked in the 3 carbs that fouled the plugs first, but only partially blocked in the last one. In my book, in the Carbs section, I show 2 pictures where I am pushing a soft mechanic's wire in thru this convoluted passage: it starts with a dogleg right at the little brass port in the air bell at the back of the carb, and then they go straight to the mainjet emulsifier. You must remove the needle jet (gently, please) in the middle of the carb's throat to push the wire thru to there, and 'saw' it back-and-forth a few times to push the corrosion out thru the emulsifier's passage. It will be a dull grey color. Then flush well (I use Brake Cleaner spray, with goggle and gloves and a shirt I don't wish to keep...) and reassemble. Do it to all 4 carbs: if one or more have this, they all do.
Later on I discovered this happens when MTBE fuel was used and allowed to evaporate more than once from those passages. It left behind a white zinc powder that was dissolved from the casting itself around the passage. Still later, someone who had access to the tiny brass plugs that you see at the back of the carb that plug this passage after it was straight-drilled, removed the plugs and took marvelous micro-pictures of the inside of an affected K1's carb (657A, similar in design) that had suffered the same fate.
Happily, ethanol-laced gas doesn't do this: MTBE fuels were intentioned to remove cars and bikes like these from the American hiways by damaging their zinc-aluminum carbs and intake manifolds: fortunately for us, we didn't use enough of it to kill these carbs.
