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Turn your A/C switch on, and see if the electric fans run. It's common for them to fail, since the control module is built into the left side fan assembly.
You should be able to run the engine to temp, if the thermostat is opening the upper hose will get hot while the lower hose stays cold, when the thermostat opens then the lower hose will get warmer as the hot coolant going into the radiator is cooled (slightly) by the fans running, and is slightly cooler going back to the engine than what is coming out. You can check it with your hands, but it's best to have one of those handheld laser thermometers (they sell them at Home Depot, Sears, etc)
This is how you test those two things- if the water pump is not pumping, then the hot coolant will never get to the radiator, it will just sit in the engine and eventually boil over. If the thermostat is not opening, then the coolant will be circulated (and you will get heat in the interior, which you typically would not if the coolant pump is not working) but the lower hose will stay cold no matter how hot the engine gets.
They will both be cool if the engine has not been running- but yes, the lower hose should be cooler than the top at all times.
If the thermostat is opening, it should at least be warm.
Keep in mind that any leak in the system will not allow it to build pressure, and the coolant will "boil" and the engine will overheat. So if your coolant cap is loose, or you have any leaks in the seals or hoses, that could be your issue.
Does the coolant actually boil over, or is it just the coolant temp gauge on the cluster showing it is overheating?
Which coolant temp sensor did you replace? Where is it located?
There is another Engine Coolant Temp sensor in the lower radiator hose, right by the radiator. I would replace that one as well, the two sensors work in conjunction. Also, check the connectors on BOTH sensors to make sure there is no sign of coolant migration or corrosion on the pins. If you still have the same issue, then I would suspect the thermostat, though if that is the case you should have a check engine light on or at least stored faults.
You can often have faults read for FREE at large auto parts stores like Autozone