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sounds like the fuel sensor inthe fuel tank which means removing tank to replace
I had this problem with my 2002 montana. The fuel gauge inside the tank was defective I went to a garage where and had the gauge changed (you can buy the part from the dealer ) You have to take the fuel tank down and unplug the wires and the fuel lines(you need a special tool to disconnect the fuel lines) Parts and labor cost me $260.00
you will need to replace the fuel sensor and it is all in one with the fuel pump its in the gas tank.
Ours had a bad instrument cluster. It cost around $300 to fix by replacing the whole instrument cluster. There was nothing actually wrong with the temp it was just the cluster malfunctioning. Ours usually occured in the winter and then worked fine in the summer.
The engine coolent may be low enough that sensor is not surounded with coolent when engine is not running.
Get the head gasket problem fixed.
The thermostat is probably stuck in the open position.
If your engine light comes on and your temp gauge is lower then normal. listen for cooling fans. if they are also running chances are you need a new thermostat
Try to locate the temperature sending unit on the engine and make sure its not a loose wire to the sending unit,if the wire is in place then the sending unit is shot.
it could be a range of things. it could be overheating, gas gauge is low, or your rpms are running too high.
Go to a Pontiac Mechanic and spend a few bucks.
Because it is a Pontiac. Sorry, man.