Recovering email

Whenever I run out of storage, outlook seem to think I've been hacked even though no-one has had access to any of my emails at all.

It's now three separate email addresses this has happened too and keep failing the online verification due to how many years ago I set these up they want emails I had sent etc...

So I'm looking to move to a new email provider for obvious reasons but wandered if there's anyway to save some important emails/documents I've now lost due too outlook. I still know my password to all 3 accounts.