iCloud / Backup

Hello Apple Community!

I am trying to backup pictures and information from my phone onto computer before I go into the Apple store to get my phone fixed (as stated in the 'before you come in' after you set up an appointment). However, when I attempt to back it up onto the computer it fails because my "iCloud storage is all used up". My computer has plenty of space to add more data so I figured it would be no issue. Is it just the storage on my phone then? (I have around 7K photos and videos on a 256 GB iPhone X). So in a nutshell - what do I do? Do I purchase more storage than the free 5 GB on my phone? I have had my phone for a little over a year and I have so many pictures I don't want to lose if they do end up having to trade in or replace my broken phone (or maybe they will just fix my broken screen and I have nothing to worry about in the first place). Anyways, I just want to make sure I do the right thing before I go in tomorrow so any help or advice would be greatly appreciated!

Thank you,

A confused college kid.