I had to do the same thing, but for a different reason.
What I did was change the SMTP address of the mailboxes.
Example:
BigBox@example.com < This box is filled with clutter
NewShinyBox@example.com < Empty
I created the "NewShinyBox@example.com" mailbox and smtp-address on the Exchange server. At the time I was ready for switching, I removed the "BigBox@example.com" SMTP address from the old cluttered mailbox, and added a new one like "BigBox_old@example.com". On the new mailbox, I added the SMTP address "BigBox@example.com" as an extra SMTP address.
All new incoming mail would be routed to the new mailbox, while the old mailbox would still be alive and operational. It will not recieve any new emails, but you will be able to access it like any other mailbox. This would give you time to clean it out and copy the emails you want to keep before removing the box itself.
If needed and if you wish (I did this), set the SMTP PRIMARY address of the new mailbox to the old primary address (in this example "BigBox@example.com").
Edit: When doing this, it will take some time to update your internal address book. Mail from outside of the organization, will end up in the correct place - but internal mail can use a bit longer to reach the correct box; but atleast you're on the correct path!