two versions of carbs were used on that model, Primer and Autochoke.. Do you have the little red bulb you push on the front of the mower?
First we probably ought to rule out, or find out, if it's a fuel issue.
when it dies and you start it back up, and it's idling real low, i would look and see what position the throttle is in, and try to manual change the RPM by adding tension to governor spring or opening the throttle by hand on the carburetor.
if it seems to run better with throttle wide open (more fuel can be drawn in) i would suspect a fuel delivery issue. if runs worse or just dies when you open the throttle, it may be the valve lash is too tight, and then when they get hot and expand there is very little to no clearance and they may not open at the precise time. i think the lash specs are 005" to 007" intake and 007" to 009" exhaust.