Nick, The only thing on the bottom of the carb should be the drain plug and the overflow. That is a drain plug, with an O-ring that seals the bowl. Take a look at the attached image and see if that is what leaks. The part number for that is 13278-02340 and it is readily available. The fuelcock assembly should not be supplying fuel in all positions. It is possible that the membrane/gasket on the inside is leaking by. I definitely would replace it. The part number for that is 44300-02262. I suggest to download the service manual for your moped from here. Hope this helps, good luck.
The first image is the drain screw and the second one is the overflow. Normally there is a drain hose attached to it. Now here comes the real issue. Why is your carb overflowing? Check your fuelcock. It is activated by engine vacuum, so there is no cut off required. Sometimes the fuelcock diaphragm will wear out and allow fuel to flow to the carburetor when the unit is shut down, which will cause excess fuel to dump out the overflow tube. Replace the fuelcock would be the way to go. Also, make sure that the fuel lines are properly routed. You only use the Pri to prime the carb, to let some gas into the bowl. When you want to start the engine you switch it to on. Now lets say that it is hooked up right, fuel flows into the float bowl from the gas tank. After the fuel reaches a certain level, it elevates the float and closes the needle valve though which the fuel flows. If something interferes with this, the gas level will continue to rise and will overflow out of the overflow tube. It could be a few of these like either A) your float drain valve is open, or B) your float height is wrong or C) your float needle is messed up. you may just be do for a quick carb rebuilt :-) Hope this helps out. Good Luck.