Moving folders and files from a classic sharepoint site to a new team site.

I would highly recommend looking at the SharePoint migration tool: https://docs.microsoft.com/en-us/sharepointmigration/introducing-the-sharepoint-migration-tool

Or some other 3rd party solution to move content.

If you move the content manually, your name is going to be all over the created by / modified by etc. Using a tool to migrate it will move those files intact as is and in bulk. You also do not have to deal with browser / trusted sites etc.

You do need to make sure you are at least the site collection admin of the sites, you don't have to be owner or added to the groups to move content to sharepoint. You can script that by doing the following on the sites:

To add access to Site collection
Connect-SPOService -Url https://tenant-admin.sharepoint.com -credential username@domain.org
Set-SPOUser -Site https://tenant.sharepoint.com/sites/sitename -LoginName username@domain.org -IsSiteCollectionAdmin $true