Wanting to switch DE with reinstalling but I can't backup my files to a flash drive

OK … so whats the issue ?

Why? How?

Maybe show us

lsblk -mp | sed -e "s/$HOSTNAME/HOST/gI" -e "s/$USER/USER/gI"

(edit - added pipe to hide possible identifiers)

PS. Just to jump the gun … I am guessing is something like this:
Root is RW, but all other drives are RO - #2 by cscs