If the temperature climbs only on grades and drops back on level ground, the problem is usually heat rejection: the fan viscous clutch, a clogged radiator, the intercooler. Sensors and coolant level read normal, which is why the warning light comes on late — once the margin is already used up.
The engine is already past its temperature limit. An aluminium head goes fast: first the geometry warps, then the gasket burns through.
Next comes a blown head gasket, a damaged head and, at worst, a cracked block. Shut it down and let it cool — never open the expansion tank cap while hot.
Let the engine cool — never open the expansion tank cap while hot. Then check the level and the hoses.
The answer depends on the code, not the light. A yellow light is a warning; a red one means do not move. But even under a yellow light there are faults that require stopping right away — low oil pressure, overheating, lost brake pressure. Enter the code — the reference will tell you plainly whether to keep driving or stop.
Know the code number? Type it into the search — or paste the whole list from the scanner and we will show which fault is the real one.