Do you still have the orange Dexcool in it?
Pull the oil cap off and look for signs of coolant in the oil (will look muddy brown/yellow). If you see this stop driving it and replace the LIM gaskets before you toast the motor.
If you don't see any signs of coolant in the oil then I would suspect a leaking radiator or hose. Since the side tanks on these are plastic they will leak eventually. Went through this on my GT. Mine cracked around the upper hose on the drivers side neck. Would only leak when hot and took forever to find.
That temp sounds about right as I believe the GTP's had a 195 T-stat in them stock.
Could also be a bad coolant level sensor. This is located on the passenger side of the rad. under the top transmission line. Should have a square back on it with a wire connection to it. You can try removing it (yes, you will loose coolant) and cleaning it. Or just replace it altogether since its 12 years old. You may have to remove the battery to get to it, can't remember for sure. I know its in a tight space.
