Found it Dan! I traced down the cable and it is attached to a bellcrank lever, by the blower motor resistor, and goes back to the left side of the radio. If it works might be a different story.
Sounds like not from what you're describing (freeze-up). You should be able to hear it click on and off loudly. You can test it by removing it, putting a bunch of ice in a tall drinking glass and filling it up with cold water , sticking the sensor capillary tube into the ice water, and working the switch lever back and forth through its range of motion. No click = no work.
Also, my car does not have a heater control valve on the inner fender, like other cars I've seen.
Each and every '65-'76 A-body with factory air had a heater valve on the RH inner fender. Perhaps yours went away in a previous crash repair or something. The '65-'69 models used an unsprung valve with 5/8" inlet and outlet; the '70 models used an otherwise identical valve with 1/2" inlet and outlet. These valves use a double-acting vacuum pot serviced as a separate component (not included with replacement heater valve: vacuum applied to one side of the pot opens the heater valve, vacuum applied to the other side closes the valve. The '71-up models use a valve spring-loaded open, with an inbuilt single-acting vacuum pot: applying vacuum closes the valve. 1/2" in and out 'til the middle of '76 production, 5/8" in and out til the end of A-body production. The '65-'69 valve is
this one. Spendy, and you'd still have to find the double-acting vacuum pot (also spendy as NOS, tough to find as a working used part). The heater valve should be closed when A/C or Max A/C is selected, open when Heat or Defog is selected. It would not be difficult to alter the vacuum hose setup and retrofit the late -'76 spring-loaded valve so that it would work correctly on a pre-'71 application like yours and I just
happen to have one, a newer-design part that doesn't bolt to the fender (small enough and light enough weight that it just goes in the heater supply hose and hangs in space) I can't use on my '73 which has 1/2" heater supply hose, not 5/8" like your pre-'70 and the late-build '76s, brand new in box, for a great deal less money than the NOS '65-'69 valve linked above. Send me a PM if you'd like it.