Some code readers will display what various sensors are reading while the engine is running. Using such a reader, you could see what it's reading for the coolant temp and compare it to your dash gauge. It's not an exact science but if the two temperature reading were in the ballpark to one another, you could safely assume the CTS was good.
Another method would be to disconnect the plug from the PCM (engine cold) and take an ohm reading on the wires running to the CTS. Plug the PCM back in, run the engine up to temp, then turn it off, disconnect the PCM plug again, and take another ohm reading to compare the results.
Yes it's possible to have a new part go bad. It's happened to me more often than I'd care to count.