If my car is overheating, should the over flow reservoir not fill up?

I have a new water pump, coolant level is good in the reservoir, yet my car keeps running hot. If I keep the heat on, it runs at a warm, but tolerable level. However, if it is running hot, should it not push coolant back into the reservoir? I squeezed the upper rad hose, and it moved the level under the rad cap, and in the reservoir, so I don't think it is blocked......

Also, when I start driving, it gets warm, then the water pump kicks in, and it goes to a good level, then slowly heats back up to overheating, but never "kicks" back down again. Could my thermostat just be sticking?

Thank you very much for any thoughts.