2001 Buick Century Overheating

You have a partially blown head gasket. What's happening is exhaust gases are leaking past the cylinders and into the cooling system.

When the gases build up, they create an air pocket inside of you heater core and radiator at the very highest point in the system. This prevents the coolant from circulating correctly.

That's why you can't get heat without accelerating. When you hit the gas, the pump spins fast enough to force the coolant past the air pockets. But when you slow down, it becomes clogged again.

This is the same reason it will over heat at low speeds.

To double check, remove the cap to your surge tank. Turn on the motor and place your heater controls to the full-hot position. (A/C off) Then have someone rev the motor while you look into the surge tank. If you see large bubbles, then that is the exhaust gases clogging up your system.

Also, when you first start the vehicle after being cold long enough, there should be white smoke coming out of your tail pipe until the motor warms up.