What's the engine model? Check and see if you are getting 12v to the control wire at the base of the solenoid, if you are check for 12v on the starter side of solenoid.
When it shuts off the first thing I would do is check for spark on the ignition coils and second see if the fuel solenoid piston is pulling down and staying down as long as power is applied.
My thoughts go a weak ignition coil or fuel solenoid, i'd suggest when it shuts off check the plug wire for spark and see if the fuel solenoid is pulling the plunger down.
What are the moisture conditions when you mow? If the grass or ground has much moisture in it this also will contribute to plugging, and I agree with other the high lift is the way to go for bagging. Good luck.
Yes sir you do need blades if you what cut the grass instead of tearing it off, you might look for a lower lift blade if available they will hold up some what longer mowing in sandy conditions.