Temperature guage not working??

How do I test the temp guage?? In the morning after the engine has been cool overnight. I turn the key to accessory and immediately within 2 seconds the needle goes 1/3 the way and stays there. Even after driving over a mile including stop lights etc.. the needle does not move any further. It does not change position at all. The only time it changes is when I turn the car off - goes to the cold mark. Gas guage appears to work fine. The gas guage registers the amount of gas regardless of car on or off. However, cooling and condenser fans appear to work fine. So it doesn't appear to be overheating as such.  Furthermore, I have seen drips of coolant under car when I park and check under the car. Leak appears to be coming from the ??air-exchange/coolant expansion tank?? there is a small crack in it. I can understand the leak but not the needle not working at startup even before any real heat/pressure is applied. 1993 MX6 V6. Thanks for any help! you guys have been great.. I know I'll need more help as I just got this car and it needs some TLC. So I'll have more questions coming soon.