vc:
I've got the '97 service manual in front of me, and am looking at the illustrations, which are not all that great. There are two attachment points for the cable: 1) The "Temperatrue Blend Air Door Cable Attachment" at the rear of the "Heater A/C Control Panel," and 2) The cable attachment point at the left, top of the "Heater A/C Housing," which is located under, and slightly to the left of your glove box.
The illustration shows a slot-head screwdriver being inserted into a slot in front of the "locking tab" which retains the cable at the Hearter A/C Housing end. It's tough to tell from the illustration exactly where the cable attaches to the housing, or whether it's easily reachable without pulling the whole unit.
The other end of the cable that attaches to the back of the Heater A/C Control Panel is very easy to get at. You just drop the steering column, roll down the instrument panel, disconnect the cable from the the heater A/C control on the back of the control panel, under the vacuum harness connector.
Just by looking at the diagrams, the anchorage at the back of this point (Heater A/C Control end... ) appears much lighter and flimsier than at the other end, so I'd check here for a problem, first.
Hope this helps.