If the photos have been deleted, what would you be searching for? If you delete photo files from outside of LR - i.e., without LR knowing they have been deleted - LR will continue to show the thumbnails in it Library grid and will attach to each one an icon indicating that the photo file cannot be found. So what would be the point of that?
If you delete these photos, you will no longer have a backup.
I think your best option would be to get a bigger harddisk - internal or external - for all your photos.
Or move the old photos to an external disk, and re-link them inside LR.
My Lightroom Catalogue has grown rather large and probably should be partitioned at this point, but I haven't gotten around to it quite yet.
I'm not sure what you mean by partitioned. There is usually no need to use more than one (1) LR catalog file.
How many image files do you have imported into LR? In the LR Library module's left panel look under Catalog> All Pictures
What is the actual LR catalog file size? Go to menu Edit> Catalog Settings> General> Size: