Where is this dryer located? What room?
Had a friend that had a dryer located in a garage. The dryer worked find during the night, but during the day would thermal out. Turns out the garage was getting too hot and that tripped the thermal breaker on the dryer.
Make sure you have plenty of airspace around the dryer when operating.
It is never a bad idea to clean lint out of a dryer. Pay special attention to the blower fan, duct and the heating coils for lint contamination.

When you say "shut off", do you mean everything stops? Do lights go dead? Is there a hum? Are the controls totally inert?

All the visible thermal switches effect the heat, not the motor. I may have the model wrong and so be wrong about this. If everything is stopping dead, I see several things to check.

1 - clogged airways, especially in the external duct(s), but really everywhere from the lint filter to the outside vent cover.
2 - low voltage overall or a bad connection on one leg of the power supply, either at the breaker, the outlet, the line cord, or one of the cheap little push on terminals inside. Measure it at all those points but at the very least measure voltage at the motor under load.
3 - open MOTOR thermal switch, probably hidden. Immediately when the motor dies, measure for voltage at the motor terminals. if you have voltage but no motor action, there's a problem found. This is often related to low voltage, a defective motor, or just being old and tired.
4 - bad controls.