There's a few ways to do this.
From what you describe it seems you have it currently setup so that when Outlook downloads the emails, it then deletes them off the server.
Outlook has a feature where it will download and then delete the emails after xx days, check out this link
https://support.zen.co.uk/kb/Knowledgeb ... the-Server
You'll see there you can select to leave a copy on the server, but then delete them after a number of days.
Option 2, would be to use IMAP on your account. IMAP is a synchronised email system where all emails are left on the server within a folder structure. It's useful when you access the emails from a number of devices or programs. Outlook and MailWasher would share the INBOX, MailWasher when using IMAP only loads UNREAD INBOX mails, and when you moved a message from the INBOX in Outlook, into a different folder, MailWasher would no longer see it.
It also shares deletes, so delete an email from the INBOX in MailWasher, Outlook would also remove it, and vice versa.
To use IMAP you would need to add a new account to Outlook, setup as such. Typically in MailWasher all you need to do is change it under Settings >> Accounts >> Incoming >> the switch is at the top.