How to add coolant to radiator when recovery tank may be defective?

2004 Cadillac Deville 4.6 Liter V8. DIC says "Check Coolant Level". There is
no heat from the cabin heater. All prior suggestions imply Coolant Level
Sensor is defective. My coolant reservoir is full. Car will overheat quickly, on
3 to 4 mile trip indicating very low coolant level in the radiator itself. I will
certainly replace the coolant tank and sensor. But it would be more
comfortable if I can determine that it is indeed defective. What test can I
perform that will provide proof that it is indeed the problem? What should the
sensor (Thermistor) resistance be to allow coolant to flow into the radiator?
Replacing the sensor requires replacing the tank of which it is a part, a $300
to $500 repair. To spend that amount because it MAY correct the problem
doesn't make sense without positive identification of the defect. Also, I find it
impossible to add coolant directly to the radiator. What procedure will enable
me to do so? Through the tubing connected to the recovery tank? Is air
pressure preventing coolant from flowing into the radiator? To have a vehicle
costing upwards of $50,000 when new to be made inoperable  because of a
single relatively inexpensive component that is designed so as to cost many
hundreds of dollars to be replaced is an inexcusable design flaw.