727 issues and questions

Matt your checking the fluid right and no it dosen't sound like any problem with the shifter linkage. One way to tell if the clutches/bands are burned up is to smell the fluid (if it hasn't been changed recently). If it smells burned that's a good sign the clutches and/or bands are toast.

The only other thing that can cause a runaway condition between shifts is an improperly adjusted throttle linkage or as everybody calls it the kickdown/passing gear linkage. The way to adjust it is with the engine off hold the carb wide open and adjust the trans. throttle linkage so it is at maximum.