Issues accessing account after Big Sur - external harddrive?

Hi everyone, I desperately need your help.

Last night I downloaded Big Sur on my Macbook Air 13” early 2015, and I think it was unsuccessful as this popped up: (a picture I used from another user with the same problem):

after one of the options of a startup disk finally showed up (Macintosh HD) the password was incorrect and i tried all i knew. After, I tried the Command + R trick before turning it on again, my user popped up and i filled in my CORRECT password but it again said it was incorrect!! After I could try using my Apple ID but again it showed that it ‘couldn’t handle this volume’:

i am sure it is because of the low storage... but i don’t know how to go from here. I don’t believe i have a backup on Time Machine and i am scared i’m losing all my important files for uni :(

will buying an USB or something help?? As it can maybe show up as another option for external harddrive? Or is there something else i can do?