@chooch Sounds like a fuel/air issue to me. Since it engages the blades and then dies can't really be attributed to the safety switch under seat. If that was the case the blades would not engage at all.
Try checking the carburetor. Sounds like the valve controlling the flow (choke) controlling the fuel is swelling when engine heats up and restricting flow of fuel to engine. Shutting it down lets the valve cool and allows it to reopen.
While you are servicing this problem you should also check your spark plug, air filter & fuel filter to ensure you are providing the best fuel flow possible.