+1 on the side stand. The carbs are all interconnected, so if any of the carbs is leaking it will take the gas from the other carbs PLUS whatever's in the actual fuel line after the petcock.
You mentioned running it a bit before shutoff, but I think you'd be amazed how little gas that consumes.
I battled problems like this on my CB550 endlessly after stupidly running without a filter. Small particles of rust caused endless problems on the #1 and #2 carb.
Adjust the float height as good as you can, then drop the float level a bit. It will run lean, but if that solves your problem you know where to start. If that doesn't, then you have junk in the lines and need to flush out the debris.