I did it! I fixed the wobbley shelf, all by my shelf...er...self.
It wasn't difficult once I figured out a few tricks. The first trick was keeping The Toddler occupied with a notebook and some (washable) markers. The second trick I got from watching a video on Youtube. I should probably explain how the whole thing happened first.
It was early-ish morning when I started this project. Like 10 AM. So The Toddler followed me up the stairs and we went into the library. I gave her a notebook and some of my good markers and set her up at my desk, which she loved. Then I took all the removable shelves off the bookcase and set them out of the way, removed the pegs, and laid the book case face down on the floor. Now the room I was working in is probably only 10x10, so I tried to angle the bookcase so I could slide the backing in from the hallway. That didn't work. The angle was weird, no matter which direction I turned it I couldn't get enough room to slide the backing in. The doorway was too narrow for the backing to be flat, I was so frustrated. So I took The Toddler and went downstairs and while she enjoyed an episode of Bubble Guppies I called my Stepdad to see if he had any ideas (he didn't) and watched Youtube videos on how to put together Billy Bookshelves. One guy did the easiest thing. He put the bookcase on it's side. So as soon as Bubble Guppies was over we trooped back up the stairs, and I tried it. And it worked! It took a few tries because the backing wouldn't stay in the track after about the half way point so I had to figure out how to hold it and slide it at the same time but I got it to work! I used extra nails to tack it on and now I have two bookshelves again!
I took me almost two days to get all the books unpacked and organized on the shelves like I wanted but the good news is it's done! The bad news is the shelves are full and I'm probably going to have to get another one soon anyway...Or maybe that's good news. New books are always good news!