Car overheats when I accelerate too hard and levels down when in motion. Fan is running, coolant is filled and no leaks. Is this the coolant system which needs replaced or the heads?
Based on your comment I would guess first some kind of clog in the cooling system, potentially in the radiator, or deteriorated fins on your water pump.
Doesn't sound like a clogged rad to me, because above 30-40mph a lot of air will be passing through it and thus cooling the coolant. Check that your fan is working well first, and if that then go to check where all that sealant went!
there are a couple of options no one else touched on, so i'll give it a go. Before buying a radiator, I would simply try changing the thermostat. They are about $10 at the most. Make sure you install it correctly. When they get old, they can "stick" where they will inconsistently open at the correct temp. Keeping cool at speed does imply the fan isn't working properly though, because it is really only useful when idling. Another option to consider is that your sending unit is malfunctioning.