a few things,:
if you can see no signs of a leak, and no white smoke then i would say that you have air in the system. it definitely could be a thermostat. definitely do a complete flush. prestone has a good kit, you cut one of your heater lines, install this fitting, hook up a water hose, leave the radiator cap off and let it run while your are pumping water into it. make sure you are running the heater at the same time too. it comes with a cap for the fitting, cap it off when ur done and you should be good to go. this will also help get any air out of the system. but you definitely need to drain around half of the water and fill it with antifreeze. depending on what your climate is, less antifreeze for warmer climates. but you definitely need the antifreeze, it has alot of stuff in it that prevents corrosion of the engine, and foaming of the coolant(which is bad) thermostats are cheap and easy to replace, might as well replace it anyways. id suggest getting a "failsafe thermostat. they are designed so that if they fail, they fail in the open position so you dont overheat. also making sure the temperature range of the new thermostat is important, make sure its the same as what cam out. it should say on the thermostat itself.