Have a Craftsman LT1000, model# above, engine above. Starts fine cold, runs 15 minutes, dies, cools off, starts, runs 15 min, dies, etc. Replaced oil, filter, fuel filter, gas cap, air filter, spark plug, & gas. Also let gas tank drain a bit in case of something big in the line. Haven't checked carb or fuel pump yet. Assuming those 2 are good due to good start and run for the duration. Kind of don't want to dig into the carb, but may have to. Don't think it would be safety switches with this problem. Don't really know where 2 go next. Suggestions? Also pulled the shroud off and the rotor and stator look fairly rusted, but I can turn it a full cycle. Also the fuel solonoid is measuring 41 ohms, and after it's been energized for for about 15 minutes, it's pretty hot, and then measures about 48 ohms -don't know what they normally read across the coils.

Sounds like you might have a coil / armature when it dies check and see if it has spark,a inline spark tester would be best to use.

Yes, check for spark when engine quits. I agree with DG. Carb problems usually won't let the enging run that long, just until the fuel in the bowl is depleted. New fuel cap eliminates plugged vent that can mimic your problem. If your fuel solenoid has a rubber tip on it, check to see if it has loosened on its shaft as it can act as a float and shut off the fuel supply after running for awhile even if the solenoid is activated.