Here are your parts
Replacement parts for WHIRLPOOL LER5848EQ2 | AppliancePartsPros.com

Are you sure that it always stops when it hits cool down or do you just think it stops then?
If it always stops at the same place on the timer then it would point toward a timer problem.
If you set the timer to the start position, will it start up?

Have you tried different cycle types and heat settings?
For instance air only to see if anything changes.

The reason I ask is that your shut down symptoms sound like the motor is overheating and tripping it's internal thermal protect. You then have to wait for the motor to cool down and the thermal protect to reset before the unit will restart.

Try turning the drum, in the correct direction, it should turn fairly easily. You may want to try this on a couple other units to get a feel for it. You are trying to see if the drum is too hard to turn and is putting too much load on the motor.

Check that the seals in the unit close to the motor/blower are OK and they are not allowing hot air to blow onto the motor.