Rough idling and stuttering on acceleration are common maladies with older marine engines that have spent lots of hours trolling. Slow speed tends to carbon foul these engines, which makes them run rough and waste fuel. Older two-stroke outboard motors are the worst offenders. Though four-stroke outboards, inboards and stern drives can sometimes also suffer the malady. The simple solution is to:

The quick and easy remedy for restoring a smooth idle comes in an aerosol can. All engine builders market their own brand of internal engine cleaners that work similarly to oven cleaner on a combustion chamber,  then at fast idle, aim the plastic straw down the throat of the carburetor. Wash the throttle plate with spray. After a few seconds the motor will begin to bog down. Keep squirting until the motor dies.

Let the chemicals work for about a half an hour, then start the engine. If the engine is hard starting, simply remove the sparkplugs and crank the engine blowing the excess engine cleaner out of the cylinders. Then, once the engine fires, you'll see great billowing clouds of smoke emanating form the exhaust. This is normal. Soon the air will clear and the engine will run smoothly.

Point of interest, during the Second World War, B-29 bomber crew chiefs decarbonized the big 12 and 18-cyinder radial engines by slowing pouring water into the air intake (with the engines idling). The water steamed off the carbon deposits. It was easy does it, because too much water would fill the combustion chamber leaving the piston with an un-compressable mass. That would result in a bent connecting rod.