Manual valve body 727 - 3rd gear won't engage under full throttle

They should literally be in whatever gear the shifter says until it is moved to another.
Meaning a full manual should be in 3rd if you put the shifter in 3rd before you even start rolling.
You said yours shifts 2-3 late with more throttle, and earlier with less so the "full manual" and "shifts late" just don't go in the same sentence together.