We need some additional numbers here.
DYT4000 and 18.5 Briggs are too general for a specific identification. DYT4000 covers many years of production and 18.5 Briggs could be 1 of several very different engines.
Post the technical model number of the tractor (usually find that under the seat on the frame). It typically begins 917.??????
For the engine numbers, find a decal or plate on the engine that lists Model number & Type number.
The model and type numbers are the most important ones needed.

OK, i will do that when i get home this evening. I do know it was purchased new in 2004, and the it is a Briggs "intek" single cylinder 18.5HP engine. Came with a 42" deck and it is a Hydro drive.

OK, the tractor model number is 917.273643
The model# of the engine is 31P777-0348-E1

The fuel leaking out of the pump vent is not normal, but may not be related to your stuttering problem. Have you checked to see if any fuel is leaking at other times, or only when your problem shows itself? Make absolutely certain that the fuel you see is not leaking where the hose leading to the carb connects to the pump, and running down to appear that it is coming from the pump vent.

Stuttering and stalling when operating on slopes is most often caused by a defective condition in the seat switch and/or switch mounting. The first clue to look for in the scenario of a seat switch defect causing this is to pay attention to just how long it takes for the stuttering to begin after you start a pass in the right slope state. If the stuttering begins as soon as you get onto the right hand slope, its nearly certain the seat switch is the culprit. If it takes 30 seconds or longer for this to happen, it might not be a seat switch issue.
If the seat switch is mounted on the right side of the seat, that side of the seat will get lighter when the operator leans or shifts their weight to the left when on the slope. Especially likely when the seat hinges have significant wear which permits greater movement of the seat.
To test this you can try adjusting the switch so the arm or plunger gets more positive pressure against it and then mow.
You might also try to test this by deliberately shifting your weight and body position to the left side of seat while mowing on flat ground.
If it can be confirmed that the seat switch IS NOT causing the stuttering, then we will need to explore the possibility that the carb float level may be too low or that some type of solid debris (gravels??) is present in the fuel tank that obstructs the fuel supply hose on right hand slopes.

Mownie, thanks for your response.
The stuttering is not immediate, it does take at least 15-20 seconds to begin after i begin mowing on the slope. The fuel dripping out of the vent does not happen at any other time, i checked. i am sure it is coming out of the vent, as when i replaced the fuel pump i cleaned everything up pretty well, so i can see where it is coming from. If i kick off the blades. the engine recovers, assuming since the load on the engine lightens when i do that.

OK, kicking off the PTO load and getting recovery would not happen if this were a seat switch issue.
At this point I would want to check the carburetor float setting to make sure it is permitting the bowl to fill to the prescribed level.

There does not seem to be any mention (in the Briggs manual) of a float adjustment per se to be made to the carbs on the engines in this series.
Nonetheless, it is still possible that somehow the float is raising higher in the bowl (pushes the needle shut) when the carb is angled to the right and therefore tries to stop fuel from entering the bowl.
This will require careful study of the orientation of the carb and examination of what takes place (as to the float movement) when the carb is tilted to either side.
And it will need a good shot of imagination too because without the float bowl (with fuel) in place, you will just have to work out in your mind what goes on.
To that effect I will offer that it appears to be a case of where the bowl level is actually too low AT ALL TIMES but really becomes a problem when the carb is tilted to the right. At that time the bowl level gets so low that the the mixture leans out enough to cause the stuttering under a load.
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