stuff you need too be checking,
does the cooling fan cut in?
if not why, has the fuse blown, does it have power, is the fan switch not getting hot enough.
does the heater run hot?
if not why, is there air in the cooling system?
does the engine run rough when you first start it up when cold?
if so, this could be a sign of a failing head gasget.
really i expect the problem is most likly from air in the cooling system from a failing head gasget but i could be wrong.
good luck :)

Is the engine getting overheated as the light indicates? Or is it a false
light?

Jasper