Additional chat-related features and issues within Microsoft Teams for business
Hi @Krislang. I feel your pain, but you must realize that for months now, It has been published that "Saved Messages" will no longer be a part of Teams. The solution all along, was to download your existing saved messages while in the classic version. Then the debate was about how best to fill in this part of Teams that Microsoft decided NOT to include in the "New Teams".
The best solution that I have seen was posted a few days ago, and that was to "right click" the message you wish to save, and select "Create task" icon. This will save the message( in it's entirety) in your task list that is accessible from Teams and from Outlook and lets you select whether it is a Private Task or a task associated to a Teams channel, etc. You can set a due date, if you need to follow up on this by a certain time... etc.
I know that it sucks when Microsoft seemingly unneccesarily messes with working features and Changes things, just for the sake of change, but unfortunately, it is the reality we live in.
So, since it is not yet July 1st, you should still be able to switch back to the Classic Teams, and download all your saved messages, then switch to the new teams and uses Tasks going forward. That is, unless your organization moved you to the new teams and removed the ability to switch back to classic. MS will be removing Classic Teams starting July 1st, so you should hurry so as not to loose any of those saved messages.