The shifter locks only in "P" or "D".
As far as temps, those are high but what is the air temp?? City traffic or out on the open roads?
As mentioned, it's common for dirt and debris to clog the front of the A/C condensor and radiator, resulting in a loss of air flow over the coils. You can direct a stream of water from the back (fan side) to flush out dirt, paper, leaves, small animals, Honda's or whatever gets sucked up from under the car.
Corvettes are called "bottom breathers" in that cooling air (and intake air) flows under the license plate and enters the radiator. Make sure the air dam under the car is in good shape as that is needed to direct the airflow upwards.
Old radiators eventually lose some of the thickness in the metal tubes where the coolant flows simply because of the friction of the coolant flowing through them. This results in a loss of the ability to transfer heat from the coolant. Other problems can be a breakdown of the radiator hoses causing them to collapse and reduce the amount of flow. The fins in the water pump can be eaten away from corrosion and that reduces the amount of coolant passing thru the system. A sticky thermostat can cause a reduction of coolant flow too.
I would do a complete flush of the system, replace the upper and lower radiator hoses, thermostat, radiator cap, and clean the radiator core. Use a 50-50 mix of the correct coolant and distilled water.
A properly functioning cooling system should not need a lower temp thermostat but you could use a 180 degree 'stat if you stil see high temps after the above maintenance.