1996 Camry overheats only with Thermostat in

I recently replaced the radiator on my 1996 Camry 4cyl and replaced the thermostat along with it. My car then would start to overheat after driving only about 500 yards. I took the thermostat out and it ran fine with temp gauge just below midline like it used to. I then returned the thermostat and got a new one (thinking the thermostat was defective) and put the new one in boiling water and watched it open (suggestion to test it). I then put the new thermostat I tested in and my car overheats again. Is the store giving me the wrong thermostat? Also I noticed the fans never come on when I have the thermostat in even when getting hot. Should I just run my car without the thermostat?

Talked to a mechanic on the phone and he thinks that my water pump is probably bad and that it doesn't spin fast enough to keep the engine cool with the thermostat, but is spinning enough to keep it cool without the thermostat. He reccommends replacing the water pump.