"Your disk is almost full" keeps coming up despite me removing things taking up space"

You are deleting stuff you should not delete.

How to Free Up Space on The Hard Drive

1. You can remove data from your Home folder except for the /Home/Library/ folder.
2. Visit The XLab FAQs and read the FAQ on freeing up space on your hard drive.
3. Also, see Freeing space on your Mac OS X startup disk.
4. What is 'Other' and 'Purgeable' in About This Mac?
5. Files That Make Up the 'Other' Storage Category, and How to Remove Them
6. Free up storage space on your Mac.
7. See Where did my Disk Space go?.
8. Be sure to Empty the Trash to recover the space.
9. Replace the drive with a larger one. Check out OWC for drives, tutorials, and toolkits.
10. Use OmniDiskSweeper or GrandPerspective to search your drive for large files and where they are located.

Various MacBook Airs shipped with 80 GB HDD/64 GB SSD (first model) up to 512 GB SSD (max for latest model).

6 GB free is still almost full. You’ll have to pinpoint files to store elsewhere.

SSD may need 10–30% free to keep working efficiently, depending on what research you read, or who you talk to. And depending on how you use the notebook (e.g. working with large files or with many files). Also, you shouldn’t carry around all your movies, photos, or music, other documents on a small disk; make a selection and store the ‘master files’ elsewhere.