For finding the actual files from the OS, if Windows, use Everything (voidtools.com). VERY powerful, easy to use search tool. Just pick a folder or file name of something missing, and it will find it.

For OS copying/moving, try TeraCopy (freeware version sufficient), both Windows and Mac. Displays what it is doing, provides a log of what it has done, won't hang on an error, and with Verify turned on, does a checksum of each file to ensure copy/move successful.

I use both every day.

You didn't say if you're on MAC or PC. On a PC dragging and dropping copies if the destination is on a different drive but moves if the destination is the same drive. If you hold the Ctrl key while dragging windows will create a copy in the destination on the same drive. If you hold the Shift key while dragging to a different drive Windows will move the file to the different drive. A small note will appear below the cursor while dragging to indicate what is happening. I can't remember so I always check the note to make sure what is happening is what I want.

The LR back-up is just a compressed copy of the contents of LRCAT catalog file. Do not mistake that for a back up of your actual image files, which it is not. You have to back-up your image files yourself and you should be backing up the LRCAT file like all important files, to a separate disk and / or the cloud.

BTW, is the HD space issue related to the LRCAT backups? There's no automated maintenance process, so if you create a full copy of the LRCAT file every time you exit, this has the possibility of exhausting your available diskspace. You should know where these backup files reside and you should be purging the old LRCAT backup files periodically, using their datastamps. You should also reconsider your frequency where maybe 1 per week is a better approach and then once per month going into the OS and deleting all but the most recent.

I was backing up photos from our vacation. I was dragging a file folder to an external hard drive, but the drive said full.

Drag & drop is not backing up! It is moving. I guess by now you know enough to make sure you have enough storage space to move your files to. Next time , right click on your files/folder & chose copy, go to your destination location & choose paste. You don't use LR's back-up feature that is asked when exiting LR?
Sorry I can't help for your situation. I have had missing photos before choose, Find missing photos from one of the drop down menus in the library module worked for me. Good luck.