Need help on diagnosis on engin hot problem.

The car is Mazda 626, 1995 with manual transmission. After running for 10 minutes, the engine becomes hot. The temperature is not consistently very high, sometimes it goes back close to normal. I checked cooling fan a few times, the fan always works. It does not look like temperature sensor is faulty. Because when the car just starts, the indicator shows low temperature, which gradually increases as car is running. The temperature stays at normal for a while, then goes higher. I changed thermostat, but found when engine was hot, the coolant hose with thermostat was still cold. So I changed to another thermostat, and found the same thing happening. Now, I am thinking maybe water pump does not work, and coolant does not circulate. so even when thermostat opens, the hot coolant does not flow, and this hose is still cold. Does this sound right? Thanks.