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This is a classic symptom of a carburetor that has become plugged with varnish.
Yes, it can even happen on a brand new engine.
These engines are run at the factory to tune them, and they do not always get cleaned properly.
As gas gets old, it turns to varnish and clogs up the passageways inside the carburetor, not allowing enough gas to get to the engine.
This condition is cumulative. Every time gas sits, the varnish builds up just a little more, like coats of paint, until eventually gas can not flow. It will not happen overnight, but the symptoms can show up all of a sudden, even while simply stopping to refill with gas.
The use of fuel additives, such as Sta-Bil or Sea Foam will not stop this process from happening. They will greatly slow it down, but the gas will still go bad.
When this happens, either the engine simply will not start, or it will not run without the choke on (this reduces the amount of air getting pulled into the engine, changing the fuel/air mixture), or it will run but surges.
Another issue that varnish in the carb can cause is that the varnish may not allow the float needle to seal properly against the seat, causing the flow of gas to not shut off when the bowl is full. The result will be gas overflowing the carb and running into the cylinder, and possibly out the air intake. If the gas gets into the cylinder, it will seep past the rings and down into the crankcase. This will be evidenced by your oil level being over-full and/or the oil smelling like gas.
The only 2 solutions are to either replace the carburetor or give it a good, thorough cleaning.
When removing the carb, make sure to take a good picture, or make a good drawing of where all springs and linkages are attached. This will make reassembly much easier.
Most people believe that cleaning a carb involves removing the bowl and wiping it out, then spraying some carb cleaner through it.
This is simply insufficient.
To properly clean the carb, you must remove it, disassemble it (making sure to remove all non-metal parts), and soak it in a commercial solvent for several hours. Soaking it overnight is even better.
Then clean all solvent off with a spray type carb cleaner, making sure to get lots of cleaner into every hole and passage there is. Pay special attention to the tiny holes in the bore of the carb, under the throttle plate for the carbs that have these holes. Use lots of cleaner. And make sure to wear safety goggles to avoid getting the over spray into your eyes. There will be over spray.
Dry the carb with low pressure compressed air.
When reassembling the carb, make sure to use a carb kit, when one is available for your carb.
Occasionally, even a good cleaning is not going to be sufficient, and you may end up having to replace the carb anyhow. Be prepared for this.
If for some odd reason this does not help, please let me know so I can assist you further.