The most urgent step is to look in your waste basket to see if you find at least some of the deleted files. If you can undelete them from there, the links will work again.
What you are seeing when browsing your catalog are small thumbnails (max 360 x 240 pixels) stored in the thumbnail cache in the catalog folder. They are not stored as image formats (only binary objects) and the only way to recover them is to take screenshots from the organizer display.
Otherwise, the only solution is to restore from backups.
About your lack of disk space, the situation is sadly more common today with very small SSD drives on laptops. If that is the case, you'll have to consider storing your data on a different drive (external USB drive mainly). That includes your photo files library and even your catalog. As discussed in two current discussions, storing the photos in the Cloud via OneDrive or similar is not a solution.
(The scratch disk can't be on an external drive).
You'll get more targetted advice if you give some info about your hardware (menu Help >> System Info of the organizer).
I imagine the catalog has a small preview of each file. Doesn't mean you still have the file. Sounds like you need to use your backups. (You do make backups, right?) You might have to add them to the catalog again, but not if you restore to the same location.
In an organizer catalog, the 'previews' of the catalog files are stored in a sqlite database file, called 'thumb.5.cache'.
As I explained above:
"What you are seeing when browsing your catalog are small thumbnails (max 360 x 240 pixels) stored in the thumbnail cache in the catalog folder. They are not stored as image formats (only binary objects) and the only way to recover them is to take screenshots from the organizer display."
Here is a screenshot of the 'thumbnail_data_table:
As you can see, there is no image file, only a binary object. The other table shows the dimensions of the image in pixels.
The only known way to display the thumbnail is from the organizer. You can show it full screen and take a screnshot, but it will be very fuzzy due to the 360 x 240 pixel size.
So, you are right, the only hope would be to restore a backup (unless recovering from the waste basket is still available).