Paulfromnyc
Last night I took both of our phones (my wife's and mine), as well as my iPad, and scoured through all the settings. I managed to fix the problem that you have described and that I was also experiencing. It seems that, either as a result of downloading the new iOS or through operator error (I suspect the former) the settings for FaceTime were unexpectedly modified. I will admit to you that I am not an "expert" in this subject matter, but the following steps solved my problem. Do this on both phones (and any other device that is sharing your apple id). When I use the term "appropriate number" in the steps below, I mean your phone number for your phone and your wife's number when looking at her phone. Seems obvious, just wanted to be clear.
1. Select SETTINGS
2. Select PHONE
3. "My Number": Verify that the appropriate phone number is listed.
4. Go back to SETTINGS
5. Select MESSAGES
6. Under "Send and Receive" you may have several phone numbers and emails. I have my phone number, my wife's phone number, and two versions of my apple id. Make sure that only the appropriate phone number is checked.
7. Go back to SETTINGS
8. Select FACETIME
9. Under "You can be reached at Face Time At" you will see the same phone numbers and email address/apple ids as you saw in MESSAGES setting. Make sure only the appropriate phone number is checked. In my case, my wife's phone had all items selected. I unchecked everything but her number and that solved our problem.
Hope this helps you get back to normal operations.
To others, I appreciate your inputs. My wife and I have been sharing the same Apple ID since we bought our two iPhone 4's. We were assured by the Apple Geniuses that sharing the ID was okay and would work just fine. They were correct and that was long before Family Sharing came on the scene. So our Calenders, Contacts, and Notes sync on all devices (including my iPad and MBP) but our phones function separately. When we upgraded to the 5s, sharing continued to work flawlessly. Bryan your post is much appreciated, but it addresses the security aspect of sharing an ID. That is not an issue in my case or the case of Paulfromnyc. Thanks again.