TimeMachine back up (about 100GB) fails to my HD even though 900 GB available... is it because of low internal space (Macbook HD is completely full)?

Hi, my old Macbook HD is completely full and I'm trying to make a last backup and then move to the new Macbook that I just bought. However, I cannot make any backups as I get a notification "Time Machine could not back up the disk “Macintosh HD - Data” because it is nearly full."

Hence, I assume the way to bridge this issue is to save some data online for the time being, free the old Macbook of several GB and then try again (and later add the files that have been stored online temporarily to the new computer?

Pls help!