Most likely the problem is in the carburetor. Either the carburetor is partially restricted and not getting enough fuel, causing the surging and backfiring and poor running.
Or, the carburetor is bypassing too much fuel through it due to a float needle that is not seating correctly.
In either case, the solution is to clean and rebuild the carburetor. These conditions are usually caused by gasoline that is left in the carburetor for too long a period of time that has gummed things up.
Here is my standard carburetor answer that should help you.
As engines sit or get older, fuel that is left in the carburetor can turn to gum and varnish and cause this and other problems. Also, any gasoline that was left in a gas can for a period of more than 30 days must be discarded because it also has begun to turn to varnish.
Today's gasoline's contain MTBE and alcohol. (Ethanol) They turn to "Junk and garbage" very quickly. Alcohol is partially water (H2O). And they call it "Oxygenated fuels! It is the oxygen that breaks down the organic compounds in the fuel and turns the gas to "Garbage" (Gum and varnish) The fuels we had just a few years ago had no alcohol in it and would store for longer periods of time before going stale... and fuel stabilizers do almost nothing to prevent the fuel from going bad with the changes in today's fuels.
Do not buy gas from the "Discount" Stations. The discount stations get a reduced price on gas because they may be buying fuel that is nearly 30 days old already. You may be getting fuel that's nearly stale right from the pump when buying from a discount station. Purchase your fuel from the well-known stations such as Shell, BP, Sonoco, Phillips 66 etc.
More than 70% of all of our repairs in our lawn mower business are due to this same issue. You most likely have dirt, gum, varnish...etc in your carburetor plugging up the small passageways and jets in the carburetor.
The carburetor must be removed from the engine. Clean all parts with carburetor cleaner and blow out all the small holes and passageways with compressed air. Remove all of the non-metallic parts since the carburetor cleaner will cause them to be disfigured decompose and plug the carburetor as time goes on.
Wash the carburetor cleaner off of the metal parts by washing them in warm, soapy water then rinsing with clean water. Dry them by blowing it off with compressed air. Make sure that all the passageways are blown dry before reassembling.
Reassemble using a NEW carburetor rebuild kit.
DO NOT TRY TO REASSEMBLE WITHOUT USING A COMPLTE CARBURETOR KIT! You will just end up having to do the job again.
ALWAYS clean the fuel tank and replace the fuel line when doing this repair or you may have to do it all over again. The inside of the fuel line disintegrates over time and these small pieces of rubber will plug up the carburetor too. Dirt and water from a dirty fuel tank will also plug up the carburetor. Find the Model, type and serial or code numbers off of the engine and take them to your local dealer to get the carburetor repair kit.
If you do the work yourself, take pictures or at least make a drawing of where all the linkages, gaskets, and component parts go. Correct reassembly is critical.
If the carburetor still doesn't work correctly, you may have to take it to someone who has an "Ultra-sonic" cleaning machine. This machine uses carb cleaner and ultrasonic vibrations to get the very small passageways clean when traditional methods fail.
If you don't feel comfortable with this kind of repair I would suggest sending it to a reputable shop.
Sorry I'm late responding to your suggestion but I have been busy working on this machine and dealing with 2 major snowstorms here in NJ.
First of all let me say that I have done much research on this problem and I have discovered that many others are experiencing similar problems with this same unit (snowthrower modelNNN-NN-NNNNand engine model 15A114-0342-E1).
I removed carb (model 790558) and cleaned with solvent and blew out with compressed air. All passages appeared clear and no gunk in the bowl. Float and float needle looked good. I re-assembled and re-installed using original gaskets since they were still intact. I installed an inline fuel filter in the neoprene gas line, replaced spark plug and started engine using fresh fuel. Engine ran fine at idle but would start to die under load. Only way to keep it running was to set the choke one notch above the closed position.
At this point I realized that the carb appeared to be starving for fuel due to poor flow from tank. Inline filter is see-through and I noticed that it was practically empty while the engine was running with fuel barely trickling out of the outlet. I have used this type of filter before on gravity feed systems and they always fill up completely.
At one point the engine died while idling and the filter was bone dry. I pulled the gas line off the tank and fuel seemed to flow from the outlet ok. I blew compressed air into tank from the outlet side and felt no restriction. I actually removed the tank and sloshed some gas around in it and turned it upside down and drained through a coffee filter to look for dirt. Saw nothing. Is there a filter or a screen in this tank that might restrict the flow?
It's not a plugged cap vent because I ran engine with loose cap and still had the same problem.
Finally, I didn't like what I saw with the filter so I went back to a straight fuel line.
So what I have now is a machine that will sit there and idle all day but will not run well under load. I used it during this latest storm with heavy wet snow. I had to keep the choke almost closed and only use half the width of the intake. It ran but was obviously still starving for fuel and died occasionly.
Now what? Does anything I said make sense?
AFter sonsiering all of the information that you just gave me, I am convinced that there is still a passageway or two that is still partially restriced or plugged in your carburetor. This is just about the only thing that can cause the problems you are having.
There is also a very small jet hole in the bowl nut (The nut that holds the carburetor bowl on) that supplies most of the fuel to the engine for full speed operation. Take a very small wire and make sure that the jet is clear. Otherwise, you may have to replace the carburetor. (Most carbs have this, a very few have a different type of jet)
I wish I had a better answer for youm but it seams like you have done nearly everything that I would have done...unless one of the old parts or gaskets are not operationg properly...this is why I tell people to use a new COMPLETE carburetor kit when cleaning the carburetor.