That's not Scrum then. You don't add stuff mid-sprint, that's the whole point. Unless the world is literally on fire, you don't touch the sprint content.
What's your scrum master doing when you get stuff added? It's their job to prevent it.
And if work overflows, you need to spend time grooming the tasks to manageable chunks and adjust the team velocity. You can always pick up more tasks if you run out of stuff to do.