Fuel overflow while running on 2006 Bayou 250

It means the fuel shutoff device within the float bowl is not functioning. You need to clean or replace your float valve.
Also check floats to make sure they still float properly. If they leak and start filling with gas they get heavy and won't shut off the fuel supply.

Check to make sure gasoline has not made its way into your engine oil.

Well, its the only explanation I can think of for raw fuel to be coming out of the carb regardless of the location.
If you can come up with another explanation, I am all ears.

I would start by measuring the fuel service level with the carb still in place.

1) Does it use a vacuum petcock?

2) Does it have a fuel pump?

3) Is this by any chance a third party carb made in China?

If you have the old OEM carb, it may pay to rebuild it and keep the Chinese one as a backup.

Not knowing the jet sizes or needle settings or internal design of the Chinese carb makes it hard to speculate on causes. If this problem started from the moment the new carb was installed then you know the source of the problem but not the reason.

It would be worth taking the carb apart and testing float valves etc and noting the jet sizes and needle settings. Was there any other work done on your Bayou that may be relevant?

Yeah it gets difficult when facts get split between two threads.

Perhaps the issue of it only leaking while running is that the vibration is enough to cause the float valve to leak. At idle fuel demand is minimal, so if the leakage exceeds the demand it the fuel level will rise until it can exit via the port in your photo.

Have you checked fuel service level yet on this carb?

Ok, then please tell us where that port leads to and what is that port's function?