How to change a key combination without using Karabiner?
You can change the text editing bindings for Cocoa apps with DefaultKeyBinding.dict. This will work for almost all applications where the ⌘ command+← left and ⌘ command+→ right worked before. (The only way it wouldn't work is if the application creator reimplemented those shortcuts from scratch, which is doubtful as it would be much easier to just use the versions that come free with a textfield.)


If the directory ~/Library/KeyBindings/ does not exist, create it.
Create the file DefaultKeyBinding.dict in that directory with contents like this:

{
    "@\Uf702" = moveWordBackward:;
    "@\Uf703" = moveWordForward:;
    "~\Uf702" = moveToBeginningOfLine:;
    "~\Uf703" = moveToEndOfLine:;
}


The @ refers to the Command key, ~ is Option, \Uf702 is Left, and \Uf703 is Right. The selectors on the right-hand side are the same ones used in the default file, /System/Library/Frameworks/AppKit.framework/Resources/StandardKeyBinding.dict, but switched around to match what you said in your question.
Restart an application for the changes to take effect in it.


Here's an in-depth guide to the Cocoa Text System that should help explain what's going on.