Sounds like you have good contact at the clutch plate and pads. Save the adjuster screw on the clutch it self as a last resort.
I would go a step beyond what was described in just adjusting the cable.
My kit (purchased in 2010) actually came with a bad sleeve. Over the years it slowly frayed the cable until it jammed this past august and would not let the clutch engage fully.
I could get the motor started but would just spin the motor up when I gave it gas; I was going no where fast. The problem strangely first showed it self as you described: back wheel still turning when clutch lever pulled in and locked.
The burred sleeve/jacked wasn't allowing the cable to travel the distance the lever released it. I didn't notice this due to the spring in the lever that cause it to appear as though it was completely out when released.
I say this to give you an idea how much resistance the sleeve was applying to the cable. Consider the clutch arm pulling and the lever 'pushing' the cable and I still did not get consistant travel from the cable causing the clutch to slip one day and be super grabby the next. With or without cable adjustment...
When it finally frayed the cable enough that it jammed, I had to disassemble the cable. Strip away the frayed/broken strands all the way to the lug at the lever and reassemble the cable to get home where I replaced the entire cable assembly. That's when I realized the burr inside the sleeve. Replacing the cable showed me how the animal was supposed to act vs the the **** poor reliability I got from the original cable.
A cable tune up kit from walmart or a visit to a bike shop for a rear brake cable will fix this.