Cooling system problem

[looneyluden]

Hello,
I'm new to the board and have a problem. I have a '97 SCII with 44k on it. The Coolant level low and coolant light recently started flashing. I added some coolant as this had fixed the problem twice before in the last year or so. The lights don't flash anymore, but now they come on and stay on steady. And whats even weirder is the temperature gauge has since started to peg out in the red. The engine can't be overheated can it? It's got full coolant, and the coolant was flushed and replaced about 10 months ago. Even if the car has sat for hours or over-night, and I even just turn the key without starting the engine, the temp-needle will start jumping around. Every few seconds it moves up or down. This with a completely cold engine(I live in alaska)! One more symptom is that when I turn the engine off, the engine fan stays on. Also, sometimes if i just go out and turn the key without starting the engine, then turn the key all the way off again, the fan will come on.

I am not informed at all about cars. But from this info, it seems to me my car is thinking it is overheated. (coolant lights on,temp gauge in the red,fan running) But it can't be overheated because the coolant level is good and all of these symptoms happen with even a cold engine. Any ideas?