Thanks so much for your reply, Lawrence, I really appreciate it. I haven't had success, though, possibly because of a combination of the number of devices I have, the new anti-theft features on iOS, and the fact that I want to use an email address previously associated with the shared account as my new, separate Apple ID. Unfortunately, I don't know why it isn't working, because Apple, no doubt for security reasons, doesn't reveal why something doesn't work, only that it doesn't.
My main goal is to switch my iPhone over to an independent account, but I would also switch my personal MacBook and Windows laptops if I could get it to work, leaving my wife's iPhone and laptop on the old ID. I have to keep something on the old account until the new account is established, though, so I can receive auth codes.
When I attempted to sign out on my iPhone, it told me I had to turn off Find my phone, first, and imposed an hour delay. After waiting an hour, I attempted to remove the email address I wanted to use as my new Apple ID from the shared ID and it imposed another 30-minute delay. After waiting through the delays and verifying the changes had taken, I attempted to create my new Apple ID but received the message that it was unable to create the ID at this time, with no explanation as to why.
Thinking maybe it was because the phone number I entered was also the recovery phone number on the old shared account, I attempted to change it to my landline, requesting to receive the call, since my landline can't receive a text message. Although it shows as an option, the call never came, so I tried again. Still no call. Thinking maybe I needed to log out and log back in (certain other successful changes weren't reflected without doing so), I tried a third time, but still no call. My wife is out of town, but I was able to reach her and try to use her phone as the recovery phone number, but when I tried to send her the code via text message, I was told I'd already made three attempts.
Apple may not want you to have a shared account, but they sure don't make it easy to back out of one. If you have any additional insight or theories, I'd truly appreciate hearing from you again.