I followed dallalr's solution above, but I was never prompted to choose whether or not to "trust" the computer. I'm behind on my software update, so I hit "Download Only" on that screen, even though I don't have enough room to download it. It didn't start downloading but didn't give an error. In an attempt to try something new, I unchecked the "Ecrypt iPhone backup" option. Before I could even hit another button, it started backing up my phone. It actually had a pop-up window showing that it's backing it up and that it's estimating the time remaining, which hadn't happened before. It's still going, but it seems like it's actually working this time.
I'll also note that my problem was slightly different. After updating iTunes, I had the issue you described above. When I disconnected and reconnected my phone, it would then give a different error. It would tell me my computer didn't have enough storage space. I have like 400 GB free to back up a 32 GB iPhone, so clearly something was wrong. This repeated indefinitely. I tried to restart iTunes, but it wouldn't start back up.
I instead restarted my computer. iTunes then started as normal. My first attempt gave me the same original error (disconnected). I then got the same "out of space" error repeatedly, at which point I ended up here.
Hopefully this works (and works quickly) so I can upgrade to my new (working) phone! Thanks for all the help!