Shortest answer regarding your question is yes.
But since I feel like you have some misunderstanding, let me try explain.
You're fine to wait 6 days, but not because it takes 10 days for them to move away.
That 10 days thing is, when you reply "Yes, you can move out" to your villager asking you about moving out. Once you say yes, it takes 10 days for them to be boxed up.
When you say "No you can't move out" to your villager, you can just put off your town for how ever long, more than 6 days or 10 days. Once you said "No" to them asking you about moving out, no one moves away even if you load the game 10 years later.
Hope this makes sense. If not, please someone explain it better! ( as my first language is not English )