You adjust the clutch cable at the lever. It should have a lil freeplay in it. Adjust it too tight and it'll do what your saying.
IMO, I think it's your clutch basket wore. It does exactly like you explained. The clutch fibers/discs wear grooves in the basket thus the clutchs won't slide correct.
You can file/grind the grooves out but it's not advised by Zuki (I have done it).
I like to use stock parts.
Hope that helps.