I edited the speech perk tree, and the perks in it, but the old perks are still being displayed in-game.

Do I need to start a new save to see the changes?

Can someone help me please?

open creation kit, click skyrim.esm and set it active.
go to actors -> perks in the objects tree.

change what you want, save and enter a name for your esp.
use the launcher oder nexus launcher to activate your .esp.
done.

Why are you giving me instructions for something I already demonstrated I was capable of doing? You were right about the load order thought, it's fixed now.