what about water coming out of the exhaust? does the oil look like a milkshake? Is there oil drops in the coolant? Does your car idle rough? These are all signs of a blown head gasket. The broken thing is still probably at the top of the radiator. For it to go past the radiator and into the engine it would have to pass through one of the little tubes which make up your radiator.

Your car could be running hot because its getting hot outside! Mine did the same thing this time last year. It would start running warm when it was above 70 outside.Something is compromising your cooling system, just when your car needs it more.