You've described it well. Could it be something like a diaphragm in a fuel pump that works well with high rpm and doesn't do well with lower rpm? If you have a vacuum-type fuel pump, try using an auxiliary tank and hook it directly to the carb and try that.

It could also indicate a blockage or obstruction in the PILOT JET of the carb (also referred to by various other names like "slow jet", "low speed jet", "idle fuel jet", or even other unprintable names when it gives trouble). Because the pilot jet has a much smaller orifice than the main jet, the pilot jets are more apt to "catch" a small piece of trash than the main jet. Particles that can clog a pilot jet may very well pass right through the main jet. Because the pilot jet has such a tiny orifice, blobs of water can "stop up" the hole as effectively as solid debris. I personally think that if the fuel pump were at fault, or anything else that would affect "MAXIMUM fuel flow", the problem would not be what you describe (removing the load from the engine and reducing throttle to a slower position). If you had a problem related to maximum fuel flow, you would experience things like "it runs OK at idle and will throttle up to high RPM but when I engage the deck, the engines begins to slow down and run rough after a few seconds. If I disengage the deck, the engine soon speeds up again. Or "It does OK until I go up a hill (or other scenario where load on engine increases). In regard to fuel pump at idle RPM versus high RPM: Pulse vacuum fuel pumps have a "relatively flat" output volume at any given RPM. This is due to the time allowed for the pump diaphragm to "react" to the "pressure to vacuum" oscillations of crankcase pressure. At idle RPM, the pulse changes occur fairly slow, so the pump diaphragm is able to move a greater distance with each pulse, because it has enough time to move the greater distance, thus it will move a certain volume of fuel in a given PERIOD OF TIME. At higher RPM, the pulse pressure to vacuum changes occur more often (because of the greater speed of the engine pistons). At higher RPM, the fuel pump moves LESS FUEL PER PULSE, but because the pulses occur at a FASTER RATE, the net output volume of the pump (during the same time span) is about the same as at idle. The volume capacity of the pump is always engineered to be well above the actual requirements of fuel flow needed by the engine anyhow. In operation, the fuel pump will fill up the carb bowl to the "float level". When the float level is attained, the needle valve closes and no more fuel is allowed into the carb until enough fuel is consumed to let the float drop a bit, at which point, additional "make up" fuel is allowed to enter the float bowl. During the times that the needle valve is closed, the fuel pump diaphragm IS NOT MOVING, because it now has a "head of positive fuel pressure" against the diaphragm, preventing it from moving at all. This is the nature of diaphragm type pumps. Carburetted engines are always running "off of" the fuel in the carb bowl at any given moment of operation. If you are running a carburetted engine that uses a fuel pump to transfer fuel from tank to carb, and you "pinch off" the fuel hose to the carb, the engine will not react immediatly to that action (by dying). It will continue to run on the fuel already in the bowl, until it consumes that "finite volume" of fuel. Water in the carb bowl (and other places in fuel system) might be to blame for the problem you are having here. A good clean out of carb and fuel tank (and a new fuel filter) might fix it. The auxiliary tank could certainly "rule out" any concerns about fuel pump performance, but I doubt a fuel pump max flow problem because THAT would cause problems during times of HIGH fuel requirement (cutting or towing heavy) and not when the engine load becomes "lighter".

Thanks for the replies and help. My mower doesn't have a fuel pump so I can rule that out. I cleaned out the carb yesterday and I'm going to try mowing tomorrow and see if it helped. The carb was clean inside and I didn't see any debris. I wouldn't think it would be an ignition problem, it definetly seems like a fuel issue.
I'll mow tomorrow and see if it runs any better.
Thanks for your time and help.

A little off the wall, but whens the last time you cleaned the cooling fins on the engine?