727 trouble ?

Start with the simple checks first, fluid level and oil color? Bright red? Is the throttle linkage (kick down linkage) hooked up to the carb linkage? If so does the throttle linkage stop when you have the throttle pushed to the wide open position? The throttle linkage should stop when the carb is @ full throttle, meaning no more movement should be available in the throttle linkage.
How does it shift under light load applications as far as harshness and RPM level? How does it shift under heavy throttle applications?
The grinding noise along with the slipping does not sound promising, there are two band adjustments on a 727, one is an external adjustment and the other band is adjusted after you drop the oil pan.
My biggest concern is the grinding, worn out band or friction discs in one of the clutches.