I am not sure if you are using IMAP or a restricted version of MAPI.
Make sure the server has not replicated your local archive or is linked to it in any way - you would not want it to be trashed when you eventually delete the messages from the server - perhaps keep a zipped copy elsewhere for safety.
If you are unable to find a more global 'remove deleted messages from server' in Preferences you could consider:
1 - web access to delete archived emails
2 - accessing the server with POP (at least temporarily) as POP has well understood methods of deleting from server. However:
1 - Make sure POP does not delete more than required
2 - POP only communicates with the server's Inbox so will only download or delete Inbox messages.
In the longer term you could consider moving away from Exchange mail servers.