The only function of the cable on a 350 is for kickdown. Just as the throttle reaches wide open, the cable should reach its full travel. Shift points on a 350 are controlled by the modulator and the governor pressure, which is generated by the speed of the driveshaft. Adjust the kickdown cable to be full forward at wide open throttle and it should function. If it doesn't, there is an internal problem with the kickdown band and/or the servo that applies it. That's overhaul territory!
What would a bulldog cross-bred with a Shi Tzu be called?
"He ain't even got a noddin' acquaintance with the truth- - -let alone a speakin' one!"