I would have to suggest some Seafoam, or in my case, I like Lucas Oil's "tuneup in a bottle" - basically, some good injector/jet cleaner. I would suggest putting some in the tank, start it up, let it run for just a few to get it circulated through, then let it sit overnight maybe.
There are also some instructions in here somewhere about adding something directly to the carbs and letting them sit overnight. You might try that if the above-mentioned doesn't work.
I agree that these things like to be run, and mine also gets "tempermental" - sputters, coughs, etc - if I let it sit for more than 3 or 4 days without running. They are also notably sensitive as far as fuel and carbs go, and I find that mine does best if I run cleaner in it about every 3 or 4 tankfulls.
If that doesn't work, nor do the other suggestions, you might be looking at something a little more involved - other issues with carbs or maybe petcock. There's lots of info in here though, so if you keep at it, with the help of the "gurus" here, you should be able to get it figured out. Whether or not you can fix it yourself or have to take it into a shop - well, that's all based on your mechanical ability, time, etc. I had to have the carbs pulled on mine, and I took it to a good shop here in town - just didn't have the time or space (live in a condo with no garage - what the hell was I thinking?!?) Lol
Good luck = )
7800 orig mi when purchased Jan 2011
Turned 11,000 mi Nov 2011
Taillight relocation kit
Chad's highway peg mounts
Otherwise, all stock
"But it's a dry heat - like the inside of an oven"