If the coolant temp sensor is suspect you need to go further than just swapping in a new sensor. Sometimes that works but with these cars the wiring for the sensor is very prone to failure. Checking the circuit is a fairly simple task that should not take more than about 20 minutes with a few common tools.
First you need some way of seeing what the ecu thinks the coolant temperature is at such as a data logger or generic scanner. If you start with a cold engine it should be within roughly 15 degrees of ambient temperature. If it checks good then wiggle the wiring around a bit to see if it drops out. If it's still good then you're done.
If the sensor is not reading correctly there are a few possibilities that could be causing your problem. Remember that these temp sensors are ntc (negative temperature coefficient) thermistors, basically that means as temperature goes up resistance goes down. High resistance will give a low temp reading and low resistance will give a high temp reading. When you have a failure it can either show a very low or a very high coolant temperature reading.
So if the ecu shows a very low coolant temperature reading (usually -40ish) this indicates an open circuit in the signal wire to the sensor, an open sensor, an unplugged sensor, an open ground wire to the sensor, or a bad ecu. If you have this condition and it's not -40 outside you can check it fairly easily.
Start by unplugging the coolant temp sensor with the key on while monitoring the temperature the ecu reads. With the sensor unplugged it should read the minimum temperature value (-40). Then using a jumper wire or paper clip, touch the signal wire to a good ground. It should read the maximum temperature value of around 300 degrees. If not then either the signal wire from the ecu is bad or the ecu itself. If it does read around 300 degrees then jumper the signal wire and the ground pin together. It should also read about 300 degrees. If it doesn't your ground wire is bad, if it does then you have just checked all of the wiring and the ecu and the only other culprit is the sensor itself.
For the other failure, if you have an extremely hot reading. With everything plugged in if your coolant temperature reads around 300 degrees the problem is either a shorted coolant temp sensor, a short to ground in the signal wire, or a faulty ecu.
To check this, first you need to unplug the coolant temp sensor. With the key on the ecu should show the minimum temperature value (-40). If it does then replace the coolant temp sensor because it's shorted. If the temp still reads high the next step is to very carefully un pin the signal wire from the ecu connector. I do recommend unhooking the battery before unplugging the ecu but that's just me.
Anyway, with the signal wire out of the ecu connector see what the coolant temperature now reads. If it is -40 then the signal wire is shorted to ground somewhere in the harness. You can either cut open the harness to find it or just run a new wire. If the coolant temperature is still reading high then the problem is in the ecu and it will need to be repaired or replaced.
I hope this helps