I'm not sure what's under the plastic cover. If you can remove the screw and remove the cover, do it. Just don't force anything and break the cover.
I'd leave just the wire cable attached to the lever. Put the housing aside. The shifter works best when there is tension in the cable, which you can supply by pulling the cable.
Spray the heck out of it, then pull the cable and shift up the range, then right back down. Up, up, up, down, down, down. Work the shifter, while pulling the cable. And with lots of lube spray. It shouldn't take very long to figure out if the shifters can be saved.