Why am I able to kill Alduin during "Alduin's Bane"

I've noticed that most people have a problem with this quest, but mainly in the fact that Alduin is unable to be damaged. My problem is different, I am able to damage Alduin, but instead of escaping to Sovengarde, when I hit with my killing strike he actually dies as if it was the final mission. The mission is still active as if he's alive, and Paarthanax repeatedly tells me to attack him even though he's already dead. I've redone multiple saves and characters yet I've been stuck on this quest since I've got the game, how can I fix it?

Depending on the state of your game, I'd say that some script or scene didn't fire properly thus allowing you to kill the actor. Try to reload a save game you did further back and see if the issue solve itself, this is sadly the only way to properly address this kind of problem.

If you don't want to deal with reload and fishing a good game state, you may want to manually set the quest stage to 220. Refer yourself to the wiki page for the quest, open the console and type setstage 220 to finish the current quest. If the next quest do not start automatically, you may need to start it manually. You may also need to resurrect Alduin.

Keep in mind that if you do make use of the console method, that won't fix whatever broke in your current save game. You may still experience weird stuff as you move forward.