The issue you are seeing is that the underlying core implementation of the webview makes and re-uses a singleton to handle the javascript/html rendering that is not exposed to the public api. In theory, this helps performance because starting/stopping that core is expensive. In practice it causes a bunch of oddities (like configurations/cache etc being shared across the webviews presented through the public api and persisting through their destruction.)
I don't think there is a way to get it to shut down.
The closest you could get would be to put your activity in another process and hope that process goes away when your activity exits.
See
https://developer.android.com/guide/topics/manifest/activity-element.htmland the android:process chunk, give that activity it's own unique process name.
This will put your activity in another process, and you will need to use RPC stuff to get data to/from it (so poking at static variables in a service or another activity won't work.)