Maybe the rough running is down to an excessively rich mixture..
One question nobody has asked - where do you fly?
If it's hot or high (high density alt), you may be into a regime where you should probably be leaning some; assuming you're full rich, you are ALREADY very rich without carb heat. ADD carb heat and you're so far rich of correct mixture it causes rough running. Maybe. You might find leaning the mixture some will resolve.
IIRC the POH suggests leaning pre-takeoff / at full power above a DA of 5000ft... but my memory ain't that good.
If in doubt, get it looked at..