It sounds like the float needle is not completely sealing up and allowing gas to overflow the normal level in the bowl. When you turn off the petcock the fuel in the bowl is being used up so the level is more like normal, so it runs better.
This is not terribly uncommon especially on older carbs. Try (carefully) removing the float to get a good look at the needle (valve). You can remove the needle and wipe it down real good, carb cleaner, etc. Just be careful if you pull the float needle tube out as they have small o-rings that can get torn and then it will constantly leak.
Beware of the cheaper aftermarket rebuild kits as the new float valves that come in them are sometimes worse then the one you are replacing.
In theory if you get the float, float needle, and complete valve system working it can not over fill the bowl with fuel.