Hi Sharon,
I am feeling like a dope because I should have said all these things at the beginning
No problem, often people don't know what's relevant to include and what isn't.
Re the "System" disk: I would not back it up; I would remove it from TM Preferences.
Since you're running a newer version of the OS, you can have Time Machine set to back up to more than one disk -- no issue there. I'd recommend:
1. Eject your current (Toshiba) backup disk and disconnect it in preparation for step 2...
2. Is there anything you want on your LG disk? If not, then connect it and erase/reformat it. This will obviously erase it, so don't do this if you have anything you want on it!
I recommend erasing/reformatting because you want to make sure the old backups are removed, since the first backup was unsuccessful, and because I don't know whether its failing was related to how it was formatted. For instructions on formatting, see: http://pondini.org/TM/5.html
The first backup should not be interrupted, so don't let your computer go to sleep during that first backup. (Another reason to remove the previous backups from the LG disk.)
Please let me know how it goes!
After you've erased the LG disk, you'd obviously want to reconnect the Toshiba disk. You can have both disks connected at all times, and TM will switch backing up to each one: Toshiba one hour, LG the next.
I appreciate your wanting to say "this helped me" or "this solved my question," but I don't know if you can since you're not the original poster....