Coolant temperature alert and fluctuating gauge?

Coolant temperate gauge on dashboard will fluctuate from mid to high, then returns to mid, then go down to low, then back to mid, then go high again. It fluctuates no matter what speed the car is driving. When the gauge goes high the “coolant temperature!” Will alert and asks that the car must be stationary with the engine running.

The engine does not seem to be overheating (no smoke and the bonnet is not hot to touch). The engine fans are working. The only other strange thing is the inside climate control / air vent heating has not been working too great recently (takes ages to reach high temp and a couple of vents don’t seem to pump out strong air even when on high).

I’ve read so many conflicting troubleshooting. Some say it could be a faulty temperature sensor. Some say it is a faulty thermostat.

I just can’t see how the engine is “overheating” if the gauge fluctuates between low/mid/ high - surely it would stay on “high” if the engine was staying overheated?