That CVK34 doesn't have an airscrew, it has a pilot screw which works in SORT OF the same way but more direct. Instead of balancing the fuel inlet using a screw to adjust the vacuum, they just use the screw to adjust the fuel orifice and inlet rate. This is because the vacuum on a 4 stroke single cylinder isn't as balanced as it is on a 2 stroke engine. There are HUGE pulsations in the inlet vacuum.
There are a few things to check though. The first off is to pull the vacuum diaphragm and check it over for ANY rips or tears. Then once you've moved on come back and reinspect to make SURE it's not ripped, torn, dry rotted, cracked, chipped, split or any other damage.
The other thing to check on the CVK 34 is the needle jet placement. I know it sounds silly but the needle jet (the little brass orifice that the needle slides down through) is only held in by the main jet tube and a REALLY loose fit at that. Sometimes, they can slip down into the float bowl if the main jet tube isn't tightened up really well and cause erratic behaviour.
Also, take some abrasive toothpaste and a Q-tip to the float needle seat. If it has any crud in it, it'll cause the carburetor to flood some and cause it run all crazy too.
I know this also sounds crazy, but change the spark plug, check the spark plug boot and coil for resistance and clean the ground terminal that bolts in under the coil. They can all mimic a rich condition with weak spark. These kawasaki's are quite finicky to weak spark masquerading as other problems!
BTW, I know.... I have a 1999 Kawasucky Prairie 400 that knows a thing or two about abuse:
