Small Engines (Lawn Mowers, etc.): B&S 3.5 HP runs rough after warmed up, fuel stabilizer, carbureter

Question
My B&S 3.5 HP seems to be running too rich after the engine has warmed up for 10-15 minutes, as if the choke was stuck or the air cleaned was clogged with dirt.  I removed the air cleaner for a quick test and it makes no difference.  I removed the spark plug and it was caked with carbon.  I installed a new spark plug and foam air filter, but after about 10 minutes we're back to running rich.  I noticed if I tip the lawn mower back on it's hind wheels at about 45 degrees, she smooths out and runs right.  This puts the spark plug at about 1:30 instead of 3 o'clock, and the carbureter on the uphill end of the gas tank.  The oil is at the full mark, I dumped the gas out of the tank, and added fresh gas bought this morning, also adding fuel stabilizer to the 2 gallon gas can.  It seems to me there must be carbureter problems of some sort.  The engine is model 90902 manufactured on Sept 1, 1999, and purchased in May of 2000.  Do I need to rebuild the carb?  Is there something else I should look at?  Thank you for any advise.

Answer
I would start with the carb since there really isn't much else that can cause a rich condition.

As for the carb, it should be a non-adjustable type which basically means you can clean it and replace gaskets/diaphragms but that's about it.  If it still runs rich then you may have to replace the carb.

Let me know what you find.
Eric