New to me B&S Vanguard 26hp V-twin vertical is flooding out at full throttle

Yeah, I was pretty happy to get it for $50 and am glad how things are going so far. I had a single 16.5HP or so that dropped a valve seat which caused the valve to break off in the cylinder. This was then forced through the head and destroyed it. The piston was dented and likely weakened. I found a used head and installed it. I set the valve lash to spec and went on with business. It ran well for a while and then the same thing happened again with the other valve. This time it busted a hole in the piston and I junked the unit and replaced it with a 16HP Kohler Command which has been excellent.

The fins on this Intek were pretty clean. I had the housing off to do the work and it looked like there wasn't any debris blocking the fins so hopefully it was kept cool. The oil was dirty and pure black but the insides of the valve covers were clean when I checked the lash (it was right on). The valve covers are always where the sludge forms because of their relatively cool temps so was happy to see that they were clean.

I got the fluid in the transaxle changed as well and have started using it to mow. There are a few other little things I need to do but overall the project of bringing my LGT 2554 back to life is completed.

This seemed to mow well but I only have one issue. The engine seems to run-on and diesel after I turn it off, sometimes with loud backfires. This also happens if I happen to raise up off my seat while mowing. I suspect the anti-backfire solenoid is giving me issues as these seem to do more harm than good from my experiences. Usually shutting down with the deck engaged will stop this but this one will actually continue to mow with the key off. Pulling the choke kills it but I hate flooding the engine with all that gas.

The engine definitely ran smoother with more use. Maybe there are deposits from the rich running that are now burning off. I did install new plugs at the same time I solved the flooding issue so assume it isn't the plugs.

Slip a spark tester on the plug and see if it is still sparking after you turn it off.
If so then the kill circuit is broken / disconnected somewhere

If no spark then try a plug 1 grade colder.

I think the issue of the backfiring and run on was related to the engine sitting and maybe deposits that formed from the rich running condition. Either way this seems to have worked itself out with simply running the mower a few hours mowing and keeping the engine under load.

Anyway, I think this thing is going to work out OK once I get a few small bugs worked out of it. Thanks.

The new lean burn engines run way too hot for air cooled engines so yes carbon buildup will be an ignition source after the ignition is turned off.
The fuel shut off solenoid should also kill the engine by starving it however if this is the case the carbon should not build up unless there is a significant oil leak.
Not uncommon to find kill wires not connected on used mowers bought from redneck anti government conspiracy no nanny state types.

The fuel shutoff solenoid appears to be molested so I think that is part of it for sure. The last time I looked these were like a $300 part so may just live with it. Either way, the run-on issue appears to have stopped after some use. There is just a nice loud backfire from time to time at shutoff. Running the engine for a while appears to have burned off some carbon.

The unit doesn't appear to be burning oil. I agree with you that modern small engines run way too lean, likely contributing to higher temps and a backfire at shut-off. For this reason I run synthetic oils in everything I own. This one has a Rotella T6 5W40 which I have had good luck with in other engines. Engines that do use a slight bit of oil usually clear up when I use this oil. It is a diesel rated synthetic so has more detergents and anti-wear additives. Most of the stuff I get is gotten with problems so I get it cheap. Often the engines have been neglected. This Rotella seems to free up sticky rings, etc.

The jets are fixed orifices and there is no adjustment besides replacing the jets. Unfortunately the available replacement options are leaner yet for high altitude operation.

Anyway, thanks again. This thing is coming together bit by bit.

If your problem is a build up of carbon fitting a tap in the fuel line and shutting off the fuel to starve the engine will go a long way to cleaning it up.
Just do not forget to turn the engine off after it stops running or you will flatten the battery.