Shape key animation applies to all animation actions, how do I only apply it to one?

Drive the shape keys

Instead of keyframing shapekeys can drive them with a custom property on the object you are animating. Change and keyframe the custom property. Keeping you to a single action.

Example image, Basis and five keys on the Cube Mesh. Five custom properties with matching name are added to Cube object. These can be keyframed, and for default scene example will become part of the "Cube" objects "CubeAction" along with transforms etc.

From Python: drive a shape key with scripting

There is a script to set up the custom properties and drivers for you.

Converting Shapekey Action to Object Action

If you already have a shape key animation, this will transfer it to object action as custom properties.

Eg the cube has an action "CubeAction" The shapekeys are part of action "Key.001Action"]

The datapath keyframed for the value of shapekey "Key 4" is

key_blocks["Key 4"].value

will convert to a fcurve on "CubeAction" with data path to custom property "Key 4"

["Key 4"]

and copy over the keyframes.

Conversion script, change name of shapekey and object animation to suit.

import bpy

context = bpy.context

sk_action = bpy.data.actions["Key.001Action"]
ob_action = bpy.data.actions["CubeAction"]

for fc in sk_action.fcurves:
    data = [0.0] * 2 * len(fc.keyframe_points)
    fc.keyframe_points.foreach_get("co", data)
    dp = fc.data_path[len('key_blocks'):-len('.value')]
    ofc = ob_action.fcurves.find(dp)
    if ofc:
        ob_action.fcurves.remove(ofc)
    ofc = ob_action.fcurves.new(dp)
    ofc.keyframe_points.add(len(fc.keyframe_points))
    ofc.keyframe_points.foreach_set("co", data)

I think you will need a different shape key animation for every action. To simplify this you can copy the shape key you already have.

It's sometimes hard to notice but you have to click the little downward pointing ">" icon. Make sure you have only the correct shape key activated by having its checkbox checked, and having the other shape keys unchecked (since the new shape key will be a combination of all the activated shape keys). Then have the Value set to the correct value (in this case 1), and then choose New Shape From Mix.