How do I move stuff to an external drive?

ENter in Time Machine as described here (via menu - if not visible, enable it into Sys Prefs). You'll have a view like the screenshot I'm attaching. You can then navigate to the files you want to move and move them to other disks on the left sidebar.

https://support.apple.com/it-it/guide/mac-help/mh11422/mac

Control (right) - click on the files in Time Machine and select "Restore XXXX to…". Then pick the external HD as the destination. This screenshot shows how saving a Photos library to another location:

Eau Rouge has been clear: unfortunately there is no other way. Get the less useful files and park them to an external HD or whatever until you can get your stuff on the next HD. But be careful: don't fill up that. Keep 40/50 Gbyte of free space fro better overall performance and no problems if need to upgrade Mac OS. Although and upgrade can be up to 12 GB, 35/45 are needed for the installation.

Drag and drop them to the external drive.

Well you never said that in the first place.

Manually browse the Time Machine drive

locate the files you want to put on the external drive and drag and drop them.