Hello and welcome to JustAnswer.com!
I've fixed dozens of these by replacing the radiator
. What happens is calcium forms on the inside of the radiator coating the metal parts but not restricting flow. Since the radiator flows just fine it shows no signs of being plugged but because of the contamination the heat from the coolant cannot properly transfer to the outside of the radiator and as a result the engine
starts to run hot and eventually overheat.
Start it up (make sure the A/C is off) and wait for it to get hot enough for the cooling fan to come on. Once the cooling fan comes on feel the air. It will most likely be hot at first but then almost immediately become luke warm. If it is like this then replace the radiator.
If the radiator was good then the air flowing across it would be really hot the whole time the fan was on.
Once you get the radiator out if you take the time to cut it in half you'll see exactly what has been happening.
Once replaced I strongly recommend that you purchase pre-mixed coolant -OR- use distilled water to dilute it (and not tap water).
Some people have reported some luck using products like CLR in the time consuming flushing process but this often takes several treatments which may or may not help. In my opinion the best option in this situation is to just replace the thing.