I have a simple blender file with a mixamo animation imported as fbx. The animation is pushed down to the NLA and I have added a new (corrective) NLA track to do some tweakings. On this new NLA track I have a new action which includes a keyframe at the first frame of the animation in which i moved the whole armature (in pose mode) along the x axis. I just want to move the whole animation on the x axis (but not the object!) but this totally destroys the animation (random bone movement).
If you look at your corrective layer's Blending mode: it is set to "Replace".
That means, whatever keyframe you put in that layer is considered as the new value for the corresponding bone.
When you are editing an action in the action editor, you can access its Blending mode in the NLA editor by selecting the top track which represents the Action Editor.
Add is usually the simplest, though sometimes you will want Multiply. Feel free to experiment.
Though be aware, you should avoid keying any parameter or controllers that you don't want to change. Especially scale.
Here's how it looks when just keying the hip bone when moves up one unit in global space: