Sorry for not posting this information prior, wasn't thinking I'll try the additional suggestions and hope one works. I do appreciate the effort by all.
Model 289707 12.5 HP(model family 28V)
Type 1179-EL
Briggs & Stratton "Power Built with cast Iron Sleeve and Synchro-Balanced." not sure if that helps any. The engine also has on it that it conforms to 1995 to 1999 California Emission Regulations, that may help with dating the engine? I also looked for a low oil sensor, couldn't seem to locate it. I have to work on the spark suggestions, there are some good ones listed. There's no fuel pump, it works on gravity and vacuum. I'm leaning more towards the ignition/spark or the valve suggestion. I guess now I'll learn how to break down a small engine!

 

I had the same trouble with mine. It would run fine till it warmed up then it would quit. I replaced the coil and it has worked fine since. My Briggs was in a craftsman and I think it was also about a 12 hp and about 20 years old.
Very easy to change out. I had never worked on one before and was back cutting grass in no time. Just take off the air shroud and you should see it by the flywheel. It has the plug wire already attached to it.

 

I hate to say this,but the problem is, its not a Honda.

 

Sounds to me like it's draining the fuel out of the bowl. I think you might have a little piece of junk in the needle and seat.

 

I have a 18 hp twin cylinder that is doing the same thing. ihave tried all of the fixes mentioned here,plus upgraded the ignition to the newer electronic style,did away with all of the saftey switches,and replaced the key switch. and it still does it. It might run fine for two hours one time,and shut down like I turned the switch off the next time. I know it's something simple,but what?